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Optimal Stopping and SwitchingProblems with Financial Applications
Zheng Wang
Submitted in partial fulfillment of the
requirements for the degree
of Doctor of Philosophy
in the Graduate School of Arts and Sciences
COLUMBIA UNIVERSITY
2017
c©2016
Zheng Wang
All Rights Reserved
ABSTRACT
Optimal Stopping and SwitchingProblems with Financial Applications
Zheng Wang
This dissertation studies a collection of problems on trading assets and deriva-
tives over finite and infinite horizons. In the first part, we analyze an optimal
switching problem with transaction costs that involves an infinite sequence
of trades. The investor’s value functions and optimal timing strategies are
derived when prices are driven by an exponential Ornstein-Uhlenbeck (XOU)
or Cox-Ingersoll-Ross (CIR) process. We compare the findings to the results
from the associated optimal double stopping problems and identify the con-
ditions under which the double stopping and switching problems admit the
same optimal entry and/or exit timing strategies.
Our results show that when prices are driven by a CIR process, optimal
strategies for the switching problems are of the classic buy-low-sell-high type.
On the other hand, under XOU price dynamics, the investor should refrain
from entering the market if the current price is very close to zero. As a result,
the continuation (waiting) region for entry is disconnected. In both models, we
provide numerical examples to illustrate the dependence of timing strategies
on model parameters.
In the second part, we study the problem of trading futures with transac-
tion costs when the underlying spot price is mean-reverting. Specifically, we
model the spot dynamics by the OU, CIR or XOU model. The futures term
structure is derived and its connection to futures price dynamics is examined.
For each futures contract, we describe the evolution of the roll yield, and com-
pute explicitly the expected roll yield. For the futures trading problem, we
incorporate the investor’s timing options to enter and exit the market, as well
as a chooser option to long or short a futures upon entry. This leads us to
formulate and solve the corresponding optimal double stopping problems to
determine the optimal trading strategies. Numerical results are presented to
illustrate the optimal entry and exit boundaries under different models. We
find that the option to choose between a long or short position induces the
investor to delay market entry, as compared to the case where the investor
pre-commits to go either long or short.
Finally, we analyze the optimal risk-averse timing to sell a risky asset.
The investor’s risk preference is described by the exponential, power or log
utility. Two stochastic models are considered for the asset price – the geomet-
ric Brownian motion (GBM) and XOU models to account for, respectively,
the trending and mean-reverting price dynamics. In all cases, we derive the
optimal thresholds and certainty equivalents to sell the asset, and compare
them across models and utilities, with emphasis on their dependence on asset
price, risk aversion, and quantity. We find that the timing option may ren-
der the investor’s value function and certainty equivalent non-concave in price
even though the utility function is concave in wealth. Numerical results are
provided to illustrate the investor’s optimal strategies and the premia associ-
ated with optimally timing to sell with different utilities under different price
dynamics.
Table of Contents
List of Figures iv
List of Tables v
1 Introduction 1
1.1 Optimal Switching under XOU and CIR Dynamics with Trans-
action Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Speculative Futures Trading under Mean Reversion . . . . . . 5
1.3 Optimal Risk Averse Timing of an Asset Sale . . . . . . . . . 8
2 Optimal Switching under XOU and CIR Dynamics with Trans-
action Costs 12
2.1 Problem Overview . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 Optimal Double Stopping Problem . . . . . . . . . . . 14
2.1.2 Optimal Switching Problem . . . . . . . . . . . . . . . 16
2.2 Summary of Analytical Results . . . . . . . . . . . . . . . . . 17
2.2.1 XOU Model . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.2 CIR model . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Methods of Solution and Proofs . . . . . . . . . . . . . . . . . 34
2.3.1 Optimal Switching Timing under the XOU Model . . . 34
2.3.2 Optimal Switching Timing under the CIR Model . . . 46
i
3 Speculative Futures Trading under Mean Reversion 48
3.1 Futures Prices and Term Structures . . . . . . . . . . . . . . . 49
3.1.1 OU and CIR Spot Models . . . . . . . . . . . . . . . . 49
3.1.2 XOU Spot Model . . . . . . . . . . . . . . . . . . . . . 53
3.2 Roll Yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.1 OU and CIR Spot Models . . . . . . . . . . . . . . . . 60
3.2.2 XOU Spot Model . . . . . . . . . . . . . . . . . . . . . 61
3.3 Optimal Timing to Trade Futures . . . . . . . . . . . . . . . . 63
3.3.1 Optimal Double Stopping Approach . . . . . . . . . . . 64
3.3.2 Variational Inequalities & Optimal Trading Strategies . 66
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4 Optimal Risk Averse Timing of an Asset Sale 75
4.1 Problem Overview . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2 Optimal Timing Strategies . . . . . . . . . . . . . . . . . . . . 79
4.2.1 The GBM Model . . . . . . . . . . . . . . . . . . . . . 79
4.2.2 The XOU Model . . . . . . . . . . . . . . . . . . . . . 84
4.3 Certainty Equivalents . . . . . . . . . . . . . . . . . . . . . . . 89
4.4 Methods of Solution and Proofs . . . . . . . . . . . . . . . . . 95
4.4.1 GBM Model . . . . . . . . . . . . . . . . . . . . . . . . 95
4.4.2 XOU Model . . . . . . . . . . . . . . . . . . . . . . . . 101
Bibliography 108
A Appendix for GBM Example 117
B Appendix for Chapter 2 119
B.1 Proof of Lemma 2.2.5 (Bounds of Jξ and V ξ) . . . . . . . . . 119
B.2 Proof of Lemma 2.2.12 (Bounds of Jχ and V χ) . . . . . . . . . 120
ii
C Appendix for Chapter 3 122
C.1 Numerical Implementation . . . . . . . . . . . . . . . . . . . . 122
iii
List of Figures
2.1 The optimal levels vs speed of mean reversion under XOU model 24
2.2 The optimal levels vs transaction cost cb under XOU model . . 25
2.3 A sample XOU path with optimal levels . . . . . . . . . . . . 27
2.4 The optimal levels vs speed of mean reversion under CIR model 34
2.5 A sample CIR path with optimal levels . . . . . . . . . . . . . 35
3.1 Calibration of VIX futures prices . . . . . . . . . . . . . . . . 57
3.2 Optimal boundaries for futures trading under the CIR model . 69
3.3 Comparison of optimal boundaries under the CIR model . . . 70
3.4 Optimal boundaries with transaction costs under OU and XOU
models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.5 The value functions P , A, and B vs spot price . . . . . . . . . 72
4.1 Value functions and utilities under the GBM model . . . . . . 82
4.2 Optimal selling thresholds under the GBM model vs quantity 82
4.3 Value functions and optimal selling thresholds under the XOU
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.4 Certainty equivalent vs price . . . . . . . . . . . . . . . . . . . 93
4.5 Liquidation premium L(z, ν) under the XOU model . . . . . . 94
iv
List of Tables
4.1 Optimal thresholds for asset sale . . . . . . . . . . . . . . . . . 89
v
Acknowledgments
I wish to express the deepest gratitude to my advisor, Professor Tim Leung,
for his mentorship. This dissertation would not be possible without Professor
Leung’s constant support and guidance over the past five years. Pursuing
a doctoral degree is a challenging endeavor and Professor Leung was beside
me every step of the way. His passion, confidence and optimism has greatly
inspired me. I am very fortunate to be able to work with him.
I am grateful for the wonderful opportunity to study at the Department
of Industrial Engineering and Operations Research at Columbia University. I
worked alongside exceptional academic researchers and forged friendships with
some of the most brilliant people whom I have had the privilege of knowing.
The courses I have taken at Columbia enriched my education and equipped
me with the skills and knowledge to undertake doctoral research. I have also
had the pleasure of serving as a teaching assistant for many graduate courses.
I thank Professor David Yao, Professor Donald Goldfarb, Professor Cliff Stein,
Professor Tim Leung, Professor Ioannis Karatzas, Professor Mark Brown, Pro-
fessor Karl Sigman and Professor Xuedong He for these valuable experiences
that will benefit me throughout my future career. I would like to express
my great appreciations to Professor Karl Sigman, Professor Jose Blanchet,
Professor Agostino Capponi and Professor Yuchong Zhang for serving on my
dissertation committee and taking the time to peruse my thesis. I am also
grateful to Professor Jay Sethuraman, director of the PhD program, for his
support and assistance.
vi
The doctoral students at the Department of Industrial Engineering and
Operations Research form a close-knit family, my friends here have given me
countless happy memories that I will greatly treasure. For this I thank Jing
Guo, Cun Mu, Chun Ye, Linan Yang, Anran Li, Itai Feigenbaum, Di Xiao, Fei
He, Ningyuan Chen, Yin Lu, Bo Huang, Xinyun Chen, Juan Li, Chun Wang,
Jing Dong, Yupeng Chen, Chen Chen, Zhen Qiu, Shuheng Zheng, Marco San-
toli and Jinbeom Kim. I also wish to make special mention of Jiao Li from
the Department of Applied Physics and Applied Mathematics for making sig-
nificant contributions to Chapter 3.
Most importantly, I wish to thank my parents Yiping Wang and Huanqun
Zheng, my sister Kexin Wang and my wife Xin Li. My parents have provided
me with unconditional support over the years and have nurtured me into the
person I am today. I am thankful to my sixteen year old sister Kexin Wang
for giving me much laughter and joy. I hope my life experience can be a useful
guide for her future pursuits. I am deeply grateful to my wife Xin Li who has
weathered both rain and shine with me. I could not have asked for a better
life companion and I wish to share this accomplishment with her.
vii
To my parents Yiping Wang
and Huanqun Zheng,
my sister Kexin Wang
and my wife Xin Li
viii
CHAPTER 1. INTRODUCTION 1
Chapter 1
Introduction
Many trading and investment decisions involve choosing when to buy and/or
sell an asset. Concomitant to this central problem are the following questions:
1. How is optimality defined? 2. How does price dynamics affect the optimal
strategies? 3. What happens when one optimizes over an infinite sequence
of trading decisions? 4. How to trade derivatives over a finite horizon? 5.
How does the investor’s optimal trading strategies depend on his/her risk
preference?
In this dissertation, we seek to address these questions using optimal stop-
ping theory. In the subsequent sections, we describe the motivation and
methodology for each chapter in this thesis.
1.1 Optimal Switching under XOU and CIR
Dynamics with Transaction Costs
The fluctuation of prices about constant values, a phenomenon known as mean
reversion, has been witnessed in many assets. Evidence of mean-reverting be-
haviors have been identified in equities (see Poterba and Summers [1988];
Malliaropulos and Priestley [1999]; Balvers et al. [2000]; Gropp [2004]), for-
CHAPTER 1. INTRODUCTION 2
eign exchange rates (see Engel and Hamilton [1989]; Anthony and MacDonald
[1998]; Larsen and Sørensen [2007]), commodities (see Schwartz [1997]) and
volatility indices (see Metcalf and Hassett [1995], Bessembinder et al. [1995],
Casassus and Collin-Dufresne [2005], and references therein).
Schwartz [1997] proposed the XOU process as a way to model commodity
prices. Since then, the XOU model has become a favored choice for modeling
mean-reverting price processes due to its analytic tractability. Moreover, it
serves as the foundation for more complex mean-reverting models. Another
popular choice for modeling mean reversion is the CIR process, it is first in-
troduced in Cox et al. [1985] as an extension of the Vasicek (OU) model and
is commonly used to model short interest rates. The CIR process has also
been used to model volatility (see Heston [1993]), energy prices (see Ribeiro
and Hodges [2004]) and as a model for project values in real options litera-
ture. For example, Carmona and Leon [2007] study an irreversible investment
problem where the interest rate follows a CIR process. In Ewald and Wang
[2010], the authors model the project value as a CIR process and solve a Dixit
and Pindyck type investment problem.
In the first part of this thesis, we consider an optimal switching problem
where the investor is assumed to alternate between market entry and exit
actions for an infinite number of times. This is in contrast to optimal double
stopping problems where the investor cannot re-enter the market after exit.
The optimal entry and exit timings will depend on the asset price dynamics,
in particular if the price process is a super/sub-martingale, then the problem
is greatly simplified and it is optimal to either exit the market immediately or
hold the asset forever. For example, this happens in the case of an underlying
asset with a geometric Brownian motion (GBM) price process (see Example
A and Shiryaev et al. [2008]). On the other hand, as a result of the signature
swinging movements, assets which demonstrate mean reversion are natural
CHAPTER 1. INTRODUCTION 3
candidates for carrying out buy-low-sell-high investment strategies. In Chapter
2, we study the optimal timing of trades subject to fixed transaction costs
under the XOU or CIR model. Building on earlier studies of double stopping
problems (see Leung et al. [2015] and Leung et al. [2014]), we use the optimal
structures of the buy/sell/wait regions to infer a similar solution structure
for the optimal switching problems and verify using variational inequalities.
Subsequently, we compare the solution of the optimal switching problem to
that of the optimal double stopping problem.
For the XOU model, it is optimal to sell when the asset price is sufficiently
high under both double stopping and switching scenarios. However, the exact
values of the upper thresholds differ for the two problems. When it comes
to market entry, we observe that under some circumstances, it is optimal to
never enter. In this case, the optimal switching problem for an investor with a
long position degenerates into an optimal stopping problem. In the alternate
case, it is optimal under both optimal double stopping and optimal switching
problems for the investor to only enter the market when the asset price first
descends into a band above zero. Precisely, the continuation region for entry
takes the form (0, A)∪ (B,+∞), with critical price levels A and B (for details
refer to Theorems 2.2.4 and 2.2.7). In other words, the continuation region is
disconnected. This interesting observation is a result of fixed transaction costs.
In the presence of only proportional costs, it is always optimal to enter when
the price is low enough (see Zhang and Zhang [2008]).
For trading problems under the CIR model, we find that it is optimal to
liquidate when the asset price is high enough. As in the XOU model, the
exit levels differ for double stopping and switching problems. Moreover, for
the switching problem, we identify necessary and sufficient conditions under
which it is optimal to never enter. In this case, the optimal switching problem
for an investor with a long position is identical to an optimal single stopping
CHAPTER 1. INTRODUCTION 4
problem. Chapter 2 is built on Leung et al. [2015] and Leung et al. [2014].
In a closely related study, Zervos et al. [2013] consider an optimal switching
problem with fixed transaction costs under a class of time-homogeneous dif-
fusions which encompasses the GBM, CEV and other models. Their findings,
however, cannot be applied to the XOU model as it violates Assumption 4 of
their paper (see Remark 2.3.2 below). Specifically, their model assumptions
restrict the optimal entry region to be defined by a single critical level, whereas
we show that in the XOU model the optimal entry region is actually a band
strictly above zero.
Song et al. [2009] solve for optimal buy-low-sell-high strategies over a finite
horizon by way of a numerical stochastic approximation scheme. Song and
Zhang [2013] analyze the optimal switching problem with stop-loss under OU
price dynamics. In a similar setting, Zhang and Zhang [2008] and Kong and
Zhang [2010] study the infinite sequential buying and selling/shorting problem
under XOU price dynamics with proportional transaction costs. In contrast to
these studies, we study optimal switching problems under both XOU and CIR
models with fixed transaction costs. In particular, the optimal entry timing
under XOU model with fixed costs is characteristically different from that with
proportional costs.
In addition to the works mentioned above, portfolio construction and hedg-
ing problems involving mean reverting assets have also been studied. For
instance, Benth and Karlsen [2005] consider the optimization problem of an
investor who has to allocate his wealth between a risk-free asset and a risky
asset that follows an XOU process. Jurek and Yang [2007] analyze a finite
horizon portfolio optimization problem concerning an OU spread where risk
preferences are characterized by the power utility and Epstein-Zin recursive
utility. Chiu and Wong [2012] consider the optimal trading of co-integrated as-
sets with a mean-variance portfolio selection criterion. Tourin and Yan [2013]
CHAPTER 1. INTRODUCTION 5
derive the dynamic trading strategy for a pair of co-integrated stocks with
the objective of maximizing the expected terminal utility of wealth over a
fixed horizon. In the case of exponential utility, they simplify the associated
Hamilton-Jacobi-Bellman equation and obtain a closed-form solution.
1.2 Speculative Futures Trading under Mean
Reversion
Futures are an integral part of the universe of derivatives. In 2015, the total
volume of futures and options traded on exchanges worldwide rose by 13.5%
from 21.83 billion contracts in 2014 to 24.78 billion contracts. The largest
growth is in Asia-Pacific, where the combined volume increased by a remark-
able 33.7%. Number of futures contracts traded globally grew substantially
by 19.3% to 14.48 billion from 12.14 billion a year prior. The CME group
and the National Stock Exchange of India (NSE) are the largest futures and
options exchanges. The 2015 combined trading volume of CME group with its
subsidiary exchanges, Chicago Mercantile Exchange, Chicago Board of Trade
and New York Mercantile Exchange was 3.53 billion contracts, while NSE had
a volume of 3.03 billion contracts.1
A futures is a contract that requires the buyer to purchase (seller to sell)
a fixed quantity of an asset, such as a commodity, at a fixed price to be paid
for on a pre-specified future date. Commonly traded on exchanges, there
are futures written on various underlying assets or references, including com-
modities, interest rates, equity indices, and volatility indices. Many futures
stipulate physical delivery of the underlying asset, with notable examples of
agricultural, energy, and metal futures. However, some, like the VIX futures,
are settled in cash.
1Statistics taken from Acworth [2016].
CHAPTER 1. INTRODUCTION 6
Futures are often used as a hedging instrument, but they are also popular
among speculative investors. In fact, they are seldom traded with the intention
of holding it to maturity as less than 1% of futures traded ever reach physical
delivery.2 This motivates the question of optimal timing to trade a futures.
In chapter 3, we investigate the speculative trading of futures under mean-
reverting spot price dynamics. Mean reversion is commonly observed for the
spot price in many futures markets, ranging from commodities and interest
rates to currencies and volatility indices, as studied in many empirical studies
(see, among others, Bessembinder et al. [1995], Irwin et al. [1996], Schwartz
[1997], Casassus and Collin-Dufresne [2005], Geman [2007], Bali and Demir-
tas [2008], Wang and Daigler [2011]). For volatility futures as an example,
Grubichler and Longstaff [1996] and Zhang and Zhu [2006] model the S&P500
volatility index (VIX) by the CIR process and provide a formula for the fu-
tures price. We start by deriving the price functions and dynamics of the
futures under the OU, CIR, and XOU models. Futures prices are computed
under the risk-neutral measure, but its evolution over time is described by the
historical measure. Thus, the investor’s optimal timing to trade depends on
both measures.
Moreover, we incorporate the investor’s timing option to enter and subse-
quently exit the market. Before entering the market, the investor faces two
possible strategies: long or short a futures first, then close the position later.
In the first strategy, an investor is expected to establish the long position when
the price is sufficiently low, and then exits when the price is high. The opposite
is expected for the second strategy. In both cases, the presence of transaction
costs expands the waiting region, indicating the investor’s desire for better
prices. In addition, the waiting region expands drastically near expiry since
transaction costs discourage entry when futures is very close to maturity. Fi-
2See p.615 of Elton et al. [2009] for a discussion.
CHAPTER 1. INTRODUCTION 7
nally, the main feature of our trading problem approach is to combine these
two related problems and analyze the optimal strategy when an investor has
the freedom to choose between either a long-short or a short-long position.
Among our results, we find that when the investor has the right to choose,
she delays market entry to wait for better prices compared to the individual
standalone problems.
Our model is a variation of the theoretical arbitrage model proposed by
Dai et al. [2011], who also incorporate the timing options to enter and exit the
market, as well as the choice between opposite positions upon entry. Their
sole underlying traded process is the stochastic basis representing the differ-
ence between the index and futures values, which is modeled by a Brownian
bridge. In an earlier study, Brennan and Schwartz [1990] formulate a similar
optimal stopping problem for trading futures where the underlying basis is a
Brownian bridge. In comparison to these two models, we model directly the
spot price process, which allows for calibration of futures prices and provide
a no-arbitrage link between the (risk-neutral) pricing and (historical) trading
problems, as opposed to a priori assuming the existence of arbitrage oppor-
tunities, and modeling the basis that is neither calibrated nor shown to be
consistent with the futures curves. A similar timing strategy for pairs trading
has been studied by Cartea et al. [2015] as an extension of the buy-low-sell-high
strategy used in Leung and Li [2015].
In addition, we study the distribution and dynamics of roll yield, an im-
portant concept in futures trading. Following the literature and industry prac-
tice, we define roll yield as the difference between changes in futures price and
changes in the underlying price (see e.g. Moskowitz et al. [2012], Gorton et al.
[2013]). For traders, roll yield is a useful gauge for deciding to invest in the
spot asset or associated futures. In essence, roll yield defined herein represents
the net cost and/or benefit of owning futures over the spot asset. Therefore,
CHAPTER 1. INTRODUCTION 8
even for an investor who trades futures only, the corresponding roll yield is a
useful reference and can affect her trading decisions. Chapter 3 is based on
Leung et al. [2016].
1.3 Optimal Risk Averse Timing of an Asset
Sale
In chapter 4 we consider a risk-averse investor who seeks to sell an asset by
selecting a timing strategy that maximizes the expected utility resulting from
the sale. At any point in time, the investor can either decide to sell immedi-
ately, or wait for a potentially better opportunity in the future. Naturally, the
investor’s decision to sell should depend on the investor’s risk aversion and the
price evolution of the risky asset. To better understand their effects, we model
the investor’s risk preference by the exponential, power, or log utility. In ad-
dition, we consider two contrasting models for the asset price – the GBM and
XOU models – to account for, respectively, the trending and mean-reverting
price dynamics. The choice of multiple utilities and stochastic models allows
for a comprehensive comparison analysis of all six possible settings.
We analyze a number of optimal stopping problems faced by the investor
under different models and utilities. The investor’s value functions and the
corresponding optimal timing strategies are solved analytically. In particu-
lar, we identify the scenarios where the optimal strategies are trivial. These
arise in the GBM model with exponential and power utilities, but not with
log utility or under the XOU model with any utility. The non-trivial optimal
timing strategies are shown to be of threshold type. The optimal threshold
represents the critical price at which the investor is willing to sell the asset
and forgo future sale opportunities. In most cases, the optimal threshold is
determined from an implicit equation, though under the GBM model with log
CHAPTER 1. INTRODUCTION 9
utility the optimal threshold is explicit. Moreover, intuitively the investor’s
optimal timing strategy should depend, not only on risk aversion and price
dynamics, but also the quantity of assets to be sold simultaneously. In gen-
eral, we find that the dependence is neither linear nor explicit. Nevertheless,
under the GBM model with log utility, the optimal price to sell is inversely
proportional to quantity so that the sale will always result in the same total
revenue regardless of quantity. In contrast, under the XOU model with power
utility, the optimal threshold is independent of quantity, and thus the total
revenue scales linearly with quantity.
While all utility functions considered herein are concave, the timing option
to sell may render the investor’s value functions and certainty equivalents non-
concave in price under different models. For instance, under the GBM model
with log utility the value function can be convex in the continuation (waiting)
region and concave when the value function coincides with the utility function
for sufficiently high asset price. Under the XOU model, we observe that the
value functions are in general neither concave nor convex in price. If the
time of asset sale is pre-determined and fixed, then the value functions are
always concave. Therefore, the phenomenon of non-concavity arises due to
the timing option to sell. Mathematically, the reason lies in the fact that the
value functions are constructed using convex functions that are the general
solutions to the PDE associated with the underlying GBM or XOU process.
To better understand the investor’s perceived value of the risky asset with
the timing option to sell, we analyze the certainty equivalent associated with
each utility maximization problem. With analytic formulas, we illustrate the
properties of the certainty equivalents. In all cases, the certainty equivalent
dominates the current asset price, and the difference indicates the premium of
the timing option. The gap typically widens as the underlying price increases
before eventually diminishing to zero for sufficiently high price. As a conse-
CHAPTER 1. INTRODUCTION 10
quence, the certainty equivalents are in general neither concave nor convex in
price. If the optimal strategy is trivial, the certainty equivalent is simply a
linear function of asset price. Chapter 4 is adapted from Leung and Wang
[2016].
In the literature, Henderson [2007] considers a risk-averse manager with a
negative exponential utility who seeks to optimally time the investment in a
project while trading in a correlated asset as a partial hedge. Under the GBM
model, the manager’s optimal timing strategy is to either invest according to
a finite threshold or postpone indefinitely. In comparison, the investor in our
exponential utility case under the GBM model may either sell immediately or
at a finite threshold, but will never find it optimal to wait indefinitely. Evans
et al. [2008] also study a mixed stochastic control/optimal stopping problem
with the objective of determining the optimal time to sell a non-traded asset
where the investor has a power utility. In our paper, we show that the optimal
timing with power utility is either to sell immediately or wait indefinitely under
the GBM model, but the threshold-type strategy is optimal under the XOU
model.
Our study focuses on the GBM and XOU models for the asset price. A
related paper by Leung et al. [2015] analyzes optimal stopping and switching
problems under the XOU model. Their results are applicable to our case with
power utility under the same model. Other mean-reverting price models, such
as the OU model (see e.g. Ekstrom et al. [2011]) and Cox-Ingersoll-Ross (CIR)
model (see e.g. Ewald and Wang [2010]; Leung et al. [2014]), have been used
to analyze various optimal timing problems. The recent work by Ekstrom and
Vaicenavicius [2016] investigates the optimal timing to sell an asset when its
price process follows a GBM-like process with a random drift. All these studies
do not incorporate risk aversion.
Alternative risk criteria can also be used to study the asset sale timing
CHAPTER 1. INTRODUCTION 11
problem. Inspired by prospect theory, Henderson [2012] considers an S-shape
(piecewise power) utility function of gain/loss relative to the initial price. Un-
der the GBM model, the investor may find it optimal to sell at a loss. Pedersen
and Peskir [2016] solve for the optimal selling strategy under the mean-variance
risk criterion. Instead of maximizing expected utility, one can also incorporate
alternative risk penalties to the optimal timing problems. Leung and Shirai
[2015] study this problem under both GBM and XOU models with shortfall
and quadratic penalties. Other than asset sale, the problem of optimal time
to sell and/or buy derivatives by a risk-averse investor has been studied by
Henderson and Hobson [2011]; Leung and Ludkovski [2012], among others.
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 12
Chapter 2
Optimal Switching under XOU
and CIR Dynamics with
Transaction Costs
In this chapter, we analyze an optimal switching problem that involves an infi-
nite sequence of trades, when prices are driven by an XOU or CIR process. We
compare the results with earlier works on optimal double stopping problems
and identify the conditions under which the double stopping and switching
problems admit the same optimal entry and/or exit timing strategies. In the
CIR case, the investor’s optimal strategy is characterized by a lower level for
entry and an upper level for exit. However, in the XOU case, we find that the
investor enters only when the current price is in a band strictly above zero. In
other words, the continuation (waiting) region for entry is disconnected. Nu-
merical results are provided to illustrate the dependence of timing strategies
on model parameters and transaction costs.
In Section 2.1, we formulate both the optimal double stopping and optimal
switching problems. Then, we present our analytical and numerical results in
Section 2.2. The proofs of our main results are detailed in Section 2.3. Finally,
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 13
Appendix B contains the proofs for a number of lemmas.
2.1 Problem Overview
In the background, we fix a probability space (Ω,F ,P), where P is the historical
probability measure. In this section, we provide an overview of our optimal
double stopping and switching problems, which will involve an exponential
Ornstein-Uhlenbeck (XOU) process. The XOU process (ξt)t≥0 is defined by
ξt = eXt , dXt = µ(θ −Xt) dt+ σ dBt. (2.1.1)
Here, X is an OU process driven by a standard Brownian motion B, with
constant parameters µ, σ > 0, θ ∈ R. In other words, X is the log-price of the
positive XOU process ξ.
Another process considered herein is the CIR process Y that satisfies
dYt = µ(θ − Yt) dt+ σ√Yt dBt, (2.1.2)
with constants µ, θ, σ > 0. If 2µθ ≥ σ2 holds, which is often referred to as
the Feller condition (see Feller [1951]), then the level 0 is inaccessible by Y .
If the initial value Y0 > 0, then Y stays strictly positive at all times almost
surely. Nevertheless, if Y0 = 0, then Y will enter the interior of the state
space immediately and stays positive thereafter almost surely. If 2µθ < σ2,
then the level 0 is a reflecting boundary. This means that once Y reaches 0, it
immediately returns to the interior of the state space and continues to evolve.
For a detailed categorization of boundaries for diffusion processes, we refer
to Chapter 2 of Borodin and Salminen [2002] and Chapter 15 of Karlin and
Taylor [1981].
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 14
2.1.1 Optimal Double Stopping Problem
Optimal double stopping problems under XOU and CIR models have been
studied in detail in Leung et al. [2015] and Leung et al. [2014] respectively. In
this chapter, we will compare the optimal strategies of double stopping prob-
lems with that of switching problems. For convenience, we shall restate the
definitions of optimal double stopping problems under XOU and CIR models
below.
Given a risky asset with an XOU price process, we first consider the optimal
timing to sell. If the share of the asset is sold at some time τ , then the investor
will receive the value ξτ = eXτ and pay a constant transaction cost cs > 0.
Denote by F the filtration generated by B, and T the set of all F-stopping
times. To maximize the expected discounted value, the investor solves the
optimal stopping problem
V ξ(x) = supτ∈T
Exe−rτ (eXτ − cs)
, (2.1.3)
where r > 0 is the constant discount rate, and Ex· ≡ E·|X0 = x.The value function V ξ(x) represents the expected liquidation value associ-
ated with ξ. On the other hand, the current price plus the transaction cost
constitute the total cost to enter the trade. Before even holding the risky
asset, the investor can always choose the optimal timing to start the trade, or
not to enter at all. This leads us to analyze the entry timing inherent in the
trading problem. Precisely, we solve
Jξ(x) = supν∈T
Exe−rν(V ξ(Xν)− eXν − cb)
, (2.1.4)
with the constant transaction cost cb > 0 incurred at the time of purchase. In
other words, the trader seeks to maximize the expected difference between the
value function V ξ(Xν) and the current eXν , minus transaction cost cb. The
value function Jξ(x) represents the maximum expected value of the investment
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 15
opportunity in the price process ξ, with transaction costs cb and cs incurred,
respectively, at entry and exit. For our analysis, the transaction costs cb and
cs can be different. To facilitate presentation, we denote the functions
hξs(x) = ex − cs and hξb(x) = ex + cb. (2.1.5)
If it turns out that Jξ(X0) ≤ 0 for some initial value X0, then the investor
will not start to trade X (see Appendix A below). In view of the example
in Appendix A, it is important to identify the trivial cases under any given
dynamics. Under the XOU model, since supx∈R(V ξ(x) − hξb(x)) ≤ 0 implies
that Jξ(x) ≤ 0 for x ∈ R, we shall therefore focus on the case with
supx∈R
(V ξ(x)− hξb(x)) > 0, (2.1.6)
and solve for the non-trivial optimal timing strategy.
Alternatively, under the CIR price dynamics, the optimal exit and entry
problems are defined, respectively, as
V χ(y) = supτ∈T
Eye−rτhχs (Yτ )
, (2.1.7)
Jχ(y) = supν∈T
Eye−rν(V χ(Yν)− hχb (Yν))
, (2.1.8)
where
hχs (y) = y − cs, and hχb (y) = y + cb. (2.1.9)
If it turns out that Jχ(Y0) ≤ 0 for some initial value Y0, then it is optimal
not to start at all. Therefore, it is important to identify the trivial cases. Under
the CIR model, since supy∈R+(V χ(y) − hb(y)) ≤ 0 implies that Jχ(y) = 0 for
y ∈ R+, we shall therefore focus on the case with
supy∈R+
(V χ(y)− hb(y)) > 0, (2.1.10)
and solve for the non-trivial optimal timing strategy.
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 16
2.1.2 Optimal Switching Problem
Under the optimal switching approach, the investor is assumed to commit to
an infinite number of trades. The sequential trading times are modeled by the
stopping times ν1, τ1, ν2, τ2, · · · ∈ T such that
0 ≤ ν1 ≤ τ1 ≤ ν2 ≤ τ2 ≤ . . . .
A share of the risky asset is bought and sold, respectively, at times νi and τi,
i ∈ N. The investor’s optimal timing to trade would depend on the initial
position. Precisely, under the XOU model, if the investor starts with a zero
position, then the first trading decision is when to buy and the corresponding
optimal switching problem is
Jξ(x) = supΛ0
Ex
∞∑n=1
[e−rτnhξs(Xτn)− e−rνnhξb(Xνn)]
, (2.1.11)
with the set of admissible stopping times Λ0 = (ν1, τ1, ν2, τ2, . . . ), and the
reward functions hξs and hξb defined in (2.1.5). On the other hand, if the investor
is initially holding a share of the asset, then the investor first determines when
to sell and solves
V ξ(x) = supΛ1
Ex
e−rτ1hξs(Xτ1) +
∞∑n=2
[e−rτnhξs(Xτn)− e−rνnhξb(Xνn)]
,(2.1.12)
with Λ1 = (τ1, ν2, τ2, ν3, . . . ).
Under CIR price dynamics, the optimal switching problems are similarly
defined as
Jχ(y) = supΛ0
Ey
∞∑n=1
[e−rτnhχs (Yτn)− e−rνnhχb (Yνn)]
, (2.1.13)
V χ(y) = supΛ1
Ey
e−rτ1hχs (Yτ1) +
∞∑n=2
[e−rτnhχs (Yτn)− e−rνnhχb (Yνn)]
,(2.1.14)
where the reward functions hχb and hχs and are defined in (2.1.9).
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 17
In summary, the optimal double stopping and switching problems differ
in the number of trades. Observe that any strategy for the double stopping
problems (2.1.3) and (2.1.4) (resp. (2.1.7) and (2.1.8)) are also candidate
strategies for the switching problems (2.1.12) and (2.1.11) (resp. (2.1.13) and
(2.1.14))respectively. Therefore, it follows that V ξ(x) ≤ V ξ(x) (resp. V χ(y) ≤V χ(y)) and Jξ(x) ≤ Jξ(x) (resp. Jχ(y) ≤ Jχ(y)). Our objective is to derive
and compare the corresponding optimal timing strategies under these two
approaches.
2.2 Summary of Analytical Results
We first summarize our analytical results and illustrate the optimal trading
strategies. For completeness, we shall restate the main results concerning
optimal double stopping problems in Leung et al. [2015] and Leung et al. [2014]
without proof. The method of solutions and proofs of the optimal switching
problems will be discussed in Section 2.3.
2.2.1 XOU Model
We begin with the optimal stopping problems (2.1.3) and (2.1.4) under the
XOU model. Denote the infinitesimal generator of the OU process X in (2.1.1)
by
L =σ2
2
d2
dx2+ µ(θ − x)
d
dx. (2.2.1)
Recall that the classical solutions of the differential equation
Lu(x) = ru(x),
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 18
for x ∈ R, are (see e.g. p.542 of Borodin and Salminen [2002] and Prop. 2.1
of Alili et al. [2005]):
F (x) ≡ F (x; r) :=
∫ ∞0
urµ−1e
√2µ
σ2(x−θ)u−u
2
2 du,
G(x) ≡ G(x; r) :=
∫ ∞0
urµ−1e
√2µ
σ2(θ−x)u−u
2
2 du.
Direct differentiation yields that F ′(x) > 0, F ′′(x) > 0, G′(x) < 0 and G′′(x) >
0. Hence, we observe that both F (x) and G(x) are strictly positive and convex,
and they are, respectively, strictly increasing and decreasing.
Define the first passage time of X to some level κ by τκ = inft ≥ 0 : Xt =
κ. As is well known, F and G admit the probabilistic expressions (see Ito
and McKean [1965] and Rogers and Williams [2000]):
Exe−rτκ =
F (x)F (κ)
if x ≤ κ,
G(x)G(κ)
if x ≥ κ.
2.2.1.1 Optimal Double Stopping Problem
We now present the result for the optimal exit timing problem under the XOU
model. First, we obtain a bound for the value function V ξ.
Lemma 2.2.1. There exists a positive constant Kξ such that, for all x ∈ R,
0 ≤ V ξ(x) ≤ ex +Kξ.
Theorem 2.2.2. The optimal liquidation problem (2.1.3) admits the solution
V ξ(x) =
ebξ∗−csF (bξ∗)
F (x) if x < bξ∗,
ex − cs if x ≥ bξ∗,
(2.2.2)
where the optimal log-price level bξ∗ for liquidation is uniquely found from the
equation
ebF (b) = (eb − cs)F ′(b). (2.2.3)
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 19
The optimal liquidation time is given by
τ ξ∗ = inf t ≥ 0 : Xt ≥ bξ∗ = inf t ≥ 0 : ξt ≥ ebξ∗ .
We now turn to the optimal entry timing problem, and give a bound on
the value function Jξ.
Lemma 2.2.3. There exists a positive constant Kξ such that, for all x ∈ R,
0 ≤ Jξ(x) ≤ Kξ.
Theorem 2.2.4. Under the XOU model, the optimal entry timing problem
(2.1.4) admits the solution
Jξ(x) =
P ξF (x) if x ∈ (−∞, aξ∗),
V ξ(x)− (ex + cb) if x ∈ [aξ∗, dξ∗],
QξG(x) if x ∈ (dξ∗,+∞),
with the constants
P ξ =V ξ(aξ∗)− (ea
ξ∗+ cb)
F (aξ∗), Qξ =
V ξ(dξ∗)− (edξ∗
+ cb)
G(dξ∗),
and the critical levels aξ∗ and dξ∗ satisfying, respectively,
F (a)(V ξ ′(a)− ea) = F ′(a)(V ξ(a)− (ea + cb)), (2.2.4)
G(d)(V ξ ′(d)− ed) = G′(d)(V ξ(d)− (ed + cb)).
The optimal entry time is given by
νaξ∗,dξ∗ := inft ≥ 0 : Xt ∈ [aξ∗, dξ∗].
In summary, the investor should exit the market as soon as the price reaches
the upper level ebξ∗
. In contrast, the optimal entry timing is the first time that
the XOU price ξ enters the interval [eaξ∗, ed
ξ∗]. In other words, it is optimal to
wait if the current price ξt is too close to zero, i.e. if ξt < eaξ∗
. Moreover, the
interval [eaξ∗, ed
ξ∗] is contained in (0, eb
ξ∗), and thus, the continuation region
for market entry is disconnected.
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 20
2.2.1.2 Optimal Switching Problem
We now turn to the optimal switching problems defined in (2.1.11) and (2.1.12)
under the XOU model. To facilitate the presentation, we denote
fs(x) := (µθ +1
2σ2 − r)− µx+ rcse
−x,
fb(x) := (µθ +1
2σ2 − r)− µx− rcbe−x.
Applying the operator L (see (2.2.1)) to hξs and hξb (see (2.1.5)), it follows
that (L − r)hξs(x) = exfs(x) and (L − r)hξb(x) = exfb(x). Therefore, fs (resp.
fb) preserves the sign of (L − r)hξs (resp. (L − r)hξb). It can be shown that
fs(x) = 0 has a unique root, denoted by xs. However,
fb(x) = 0 (2.2.5)
may have no root, a single root, or two distinct roots, denoted by xb1 and xb2,
if they exist. The following observations will also be useful:
fs(x)
> 0 if x < xs,
< 0 if x > xs,
and fb(x)
< 0 if x ∈ (−∞, xb1) ∪ (xb2,+∞),
> 0 if x ∈ (xb1, xb2).
(2.2.6)
We first obtain bounds for the value functions Jξ and V ξ.
Lemma 2.2.5. There exists positive constants C1 and C2 such that
0 ≤ Jξ(x) ≤ C1,
0 ≤ V ξ(x) ≤ ex + C2.
The optimal switching problems have two different sets of solutions de-
pending on the problem data.
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 21
Theorem 2.2.6. The optimal switching problem (2.1.11)-(2.1.12) admits the
solution
Jξ(x) = 0, for x ∈ R, and V ξ(x) =
ebξ∗−csF (bξ∗)
F (x) if x < bξ∗,
ex − cs if x ≥ bξ∗,
(2.2.7)
where bξ∗ satisfies (2.2.3), if any of the following mutually exclusive conditions
holds:
(i) There is no root or a single root to equation (2.2.5).
(ii) There are two distinct roots to (2.2.5). Also
∃ a∗ ∈ (xb1, xb2) such that F (a∗)ea∗
= F ′(a∗)(ea∗
+ cb), (2.2.8)
and
ea∗
+ cbF (a∗)
≥ ebξ∗ − csF (bξ∗)
. (2.2.9)
(iii) There are two distinct roots to (2.2.5) but (2.2.8) does not hold.
In Theorem 2.2.6, Jξ = 0 means that it is optimal not to enter the market
at all. On the other hand, if one starts with a unit of the underlying asset, the
optimal switching problem reduces to a problem of optimal single stopping.
Indeed, the investor will never re-enter the market after exit. This is identical
to the optimal liquidation problem (2.1.3) where there is only a single (exit)
trade. The optimal strategy in this case is the same as V ξ in (2.2.2) – it is
optimal to exit the market as soon as the log-price X reaches the threshold
bξ∗.
We also address the remaining case when none of the conditions in Theorem
2.2.6 hold. As we show next, the optimal strategy will involve both entry and
exit thresholds.
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 22
Theorem 2.2.7. If there are two distinct roots to (2.2.5), xb1 and xb2, and
there exists a number a∗ ∈ (xb1, xb2) satisfying (2.2.8) such that
ea∗
+ cbF (a∗)
<ebξ∗ − csF (bξ∗)
, (2.2.10)
then the optimal switching problem (2.1.11)-(2.1.12) admits the solution
Jξ(x) =
PF (x) if x ∈ (−∞, a∗),
KF (x)− (ex + cb) if x ∈ [a∗, d∗],
QG(x) if x ∈ (d∗,+∞),
(2.2.11)
V ξ(x) =
KF (x) if x ∈ (−∞, b∗),
QG(x) + ex − cs if x ∈ [b∗,+∞),
(2.2.12)
where a∗ satisfies (2.2.8), and
P = K − ea∗
+ cbF (a∗)
,
K =ed∗G(d∗)− (ed
∗+ cb)G
′(d∗)
F ′(d∗)G(d∗)− F (d∗)G′(d∗),
Q =ed∗F (d∗)− (ed
∗+ cb)F
′(d∗)
F ′(d∗)G(d∗)− F (d∗)G′(d∗)
There exist unique critical levels d∗ and b∗ which are found from the nonlinear
system of equations:
edG(d)− (ed + cb)G′(d)
F ′(d)G(d)− F (d)G′(d)=ebG(b)− (eb − cs)G′(b)F ′(b)G(b)− F (b)G′(b)
, (2.2.13)
edF (d)− (ed + cb)F′(d)
F ′(d)G(d)− F (d)G′(d)=ebF (b)− (eb − cs)F ′(b)F ′(b)G(b)− F (b)G′(b)
. (2.2.14)
Moreover, the critical levels are such that d∗ ∈ (xb1, xb2) and b∗ > xs.
The optimal strategy in Theorem 2.2.7 is described by the stopping times
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 23
Λ∗0 = (ν∗1 , τ∗1 , ν
∗2 , τ
∗2 , . . . ), and Λ∗1 = (τ ∗1 , ν
∗2 , τ
∗2 , ν
∗3 , . . . ), with
ν∗1 = inft ≥ 0 : Xt ∈ [a∗, d∗],
τ ∗i = inft ≥ ν∗i : Xt ≥ b∗, and ν∗i+1 = inft ≥ τ ∗i : Xt ≤ d∗, for i ≥ 1.
In other words, it is optimal to buy if the price is within [ea∗, ed
∗] and then
sell when the price ξ reaches eb∗. The structure of the buy/sell regions is
similar to that in the double stopping case (see Theorems 2.2.2 and 2.2.4). In
particular, a∗ is the same as aξ∗ in Theorem 2.2.4 since the equations (2.2.4)
and (2.2.8) are equivalent. The level a∗ is only relevant to the first purchase.
Mathematically, a∗ is determined separately from d∗ and b∗. If we start with
a zero position, then it is optimal to enter if the price ξ lies in the interval
[ea∗, ed
∗]. However, on all subsequent trades, we enter as soon as the price
hits ed∗
from above (after exiting at eb∗
previously). Hence, the lower level a∗
becomes irrelevant after the first entry.
Note that the conditions that differentiate Theorems 2.2.6 and 2.2.7 are
exhaustive and mutually exclusive. If the conditions in Theorem 2.2.6 are vio-
lated, then the conditions in Theorem 2.2.7 must hold. In particular, condition
(2.2.8) in Theorem 2.2.6 holds if and only if∣∣∣∣∫ xb1
−∞Ψ(x)exfb(x)dx
∣∣∣∣ < ∫ xb2
xb1
Ψ(x)exfb(x)dx, (2.2.15)
where
Ψ(x) =2F (x)
σ2W(x), and W(x) = F ′(x)G(x)− F (x)G′(x) > 0. (2.2.16)
Inequality (2.2.15) can be numerically verified given the model inputs.
2.2.1.3 Numerical Examples
We numerically implement Theorems 2.2.2, 2.2.4, and 2.2.7, and illustrate
the associated entry/exit thresholds. In Figure 2.1 (left), the optimal entry
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 24
levels dξ∗ and d∗ rise, respectively, from 0.7425 to 0.7912 and from 0.8310 to
0.8850, as the speed of mean reversion µ increases from 0.5 to 1. On the other
hand, the critical exit levels bξ∗ and b∗ remain relatively flat over µ. As for
the critical lower level aξ∗ from the optimal double stopping problem, Figure
2.1 (right) shows that it is decreasing in µ. The same pattern holds for the
optimal switching problem since the critical lower level a∗ is identical to aξ∗,
as noted above.
0.5 0.6 0.7 0.8 0.9 1
0.7
0.8
0.9
1
1.1
1.2
µ
d∗
b∗
dξ∗
bξ∗
0.5 0.6 0.7 0.8 0.9 1−9.4
−9.2
−9
−8.8
−8.6
−8.4
µ
aξ∗
Figure 2.1: (Left) The optimal entry and exit levels vs speed of mean reversion µ.
Parameters: σ = 0.2, θ = 1, r = 0.05, cs = 0.02, cb = 0.02. (Right) The critical
lower level of entry region aξ∗ decreases monotonically from -8.4452 to -9.2258 as µ
increases from 0.5 to 1. Parameters: σ = 0.2, θ = 1, r = 0.05, cs = 0.02, cb = 0.02.
We now look at the impact of transaction cost in Figure 2.2. On the left
panel, we observe that as the transaction cost cb increases, the gap between
the optimal switching entry and exit levels, d∗ and b∗, widens. This means
that it is optimal to delay both entry and exit. Intuitively, to counter the fall
in profit margin due to an increase in transaction cost, it is necessary to buy at
a lower price and sell at a higher price to seek a wider spread. In comparison,
the exit level bξ∗ from the double stopping problem is known analytically to
be independent of the entry cost, so it stays constant as cb increases in the
figure. In contrast, the entry level dξ∗, however, decreases as cb increases but
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 25
0.02 0.04 0.06 0.08 0.1
0.7
0.8
0.9
1
1.1
1.2
cb
d∗
b∗
dξ∗
bξ∗
0.02 0.04 0.06 0.08 0.1−9.5
−9
−8.5
−8
−7.5
−7
−6.5
cb
aξ∗
Figure 2.2: (Left) The optimal entry and exit levels vs transaction cost cb. Param-
eters: µ = 0.6, σ = 0.2, θ = 1, r = 0.05, cs = 0.02. (Right) The critical lower level
of entry region aξ∗ increases monotonically from -9.4228 to -6.8305 as cb increases
from 0.01 to 0.1. Parameters: µ = 0.6, σ = 0.2, θ = 1, r = 0.05, cs = 0.02.
much less significantly than d∗. Figure 2.2 (right) shows that aξ∗, which is the
same for both the optimal double stopping and switching problems, increases
monotonically with cb.
In both Figures 2.1 and 2.2, we can see that the interval of the entry and
exit levels, (d∗, b∗), associated with the optimal switching problem lies within
the corresponding interval (dξ∗, bξ∗) from the optimal double stopping prob-
lem. Intuitively, with the intention to enter the market again upon completing
the current trade, the trader is more willing to enter/exit earlier, meaning a
narrowed waiting region.
Figure 2.3 shows a simulated path and the associated entry/exit levels.
As the path starts at ξ0 = 2.6011 > ed∗> ed
ξ∗, the investor waits to enter
until the path reaches the lower level edξ∗
(double stopping) or ed∗
(switching)
according to Theorems 2.2.4 and 2.2.7. After entry, the investor exits at the
optimal level ebξ∗
(double stopping) or eb∗
(switching). The optimal switching
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 26
thresholds imply that the investor first enters the market on day 188 where
the underlying asset price is 2.3847. In contrast, the optimal double stopping
timing yields a later entry on day 845 when the price first reaches edξ∗
= 2.1754.
As for the exit timing, under the optimal switching setting, the investor exits
the market earlier on day 268 at the price eb∗
= 2.8323. The double stopping
timing is much later on day 1160 when the price reaches ebξ∗
= 3.0988. In
addition, under the optimal switching problem, the investor executes more
trades within the same time span. As seen in the figure, the investor would
have completed two ‘round-trip’ (buy-and-sell) trades in the market before the
double stopping investor liquidates for the first time.
2.2.2 CIR model
We now turn our attention to the CIR model. We consider the optimal
starting-stopping problem followed by the optimal switching problem. First,
we denote the infinitesimal generator of Y as
Lχ =σ2y
2
d2
dy2+ µ(θ − y)
d
dy,
and consider the ordinary differential equation (ODE)
Lχu(y) = ru(y), for y ∈ R+. (2.2.17)
To present the solutions of this ODE, we define the functions
F χ(y) := M(r
µ,2µθ
σ2;2µy
σ2), and Gχ(y) := U(
r
µ,2µθ
σ2;2µy
σ2),
where
M(a, b; z) =∞∑n=0
anzn
bnn!, a0 = 1, an = a(a+ 1)(a+ 2) · · · (a+ n− 1),
U(a, b; z) =Γ(1− b)
Γ(a− b+ 1)M(a, b; z) +
Γ(b− 1)
Γ(a)z1−bM(a− b+ 1, 2− b; z)
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 27
0 200 400 600 800 1000 12001
1.5
2
2.5
3
3.5
4
4.5
Days
exp(bξ∗)
exp(b∗)exp(d∗)exp(dξ∗)
Figure 2.3: A sample XOU path, along with entry and exit levels. Under the double
stopping setting, the investor enters at νdξ∗ = inft ≥ 0 : ξt ≤ edξ∗
= 2.1754 with
dξ∗ = 0.7772, and exit at τbξ∗ = inft ≥ νdξ∗ : ξt ≥ ebξ∗ = 3.0988 with bξ∗ = 1.1310.
The optimal switching investor enters at νd∗
= inft ≥ 0 : ξt ≤ ed∗
= 2.3888 with
d∗ = 0.8708, and exit at τb∗
= inft ≥ νd∗
: ξt ≥ eb∗
= 2.8323 with b∗ = 1.0411.
The critical lower threshold of entry region is eaξ∗
= 1.264 ·10−4 with aξ∗ = −8.9760
(not shown in this figure). Parameters: µ = 0.8, σ = 0.2, θ = 1, r = 0.05, cs = 0.02,
cb = 0.02.
are the confluent hypergeometric functions of first and second kind, also called
the Kummer’s function and Tricomi’s function, respectively (see Chapter 13
of Abramowitz and Stegun [1965] and Chapter 9 of Lebedev [1972]). As is well
known (see Going-Jaeschke and Yor [2003]), F χ and Gχ are strictly positive
and, respectively, the strictly increasing and decreasing continuously differen-
tiable solutions of the ODE (2.2.17). Also, we remark that the discounted
processes (e−rtF χ(Yt))t≥0 and (e−rtGχ(Yt))t≥0 are martingales.
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 28
In addition, recall the reward functions defined in (2.1.9) and note that
(Lχ − r)hb(y)
> 0 if y < yb,
< 0 if y > yb,
(2.2.18)
and
(Lχ − r)hs(y)
> 0 if y < ys,
< 0 if y > ys,
(2.2.19)
where the critical constants yb and ys are defined by
yb :=µθ − rcbµ+ r
and ys :=µθ + rcsµ+ r
. (2.2.20)
Note that yb and ys depend on the parameters µ, θ and r, as well as cb and cs
respectively, but not σ.
2.2.2.1 Optimal Starting-Stopping Problem
We now present the results for the optimal starting-stopping problem (2.1.7)-
(2.1.8). As it turns out, the value function V χ is expressed in terms of F χ,
and Jχ in terms of V χ and Gχ. The functions F χ and Gχ also play a role in
determining the optimal starting and stopping thresholds.
First, we give a bound on the value function V χ in terms of F χ(y).
Lemma 2.2.8. There exists a positive constant Kχ such that, for all y ≥ 0,
0 ≤ V χ(y) ≤ KχF χ(y).
Theorem 2.2.9. The value function for the optimal stopping problem (2.1.7)
is given by
V χ(y) =
bχ∗−csFχ(bχ∗)
F χ(y) if y ∈ [0, bχ∗),
y − cs if y ∈ [bχ∗,+∞).
Here, the optimal stopping level bχ∗ ∈ (cs ∨ ys,∞) is found from the equation
F χ(b) = (b− cs)F χ′(b). (2.2.21)
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 29
Therefore, it is optimal to stop as soon as the process Y reaches bχ∗ from
below. The stopping level bχ∗ must also be higher than the fixed cost cs as
well as the critical level ys defined in (2.2.20).
Now we turn to the optimal starting problem. Define the reward function
hχ(y) := V χ(y)− (y + cb).
Since F χ, and thus V χ, are convex, so is hχ, we also observe that the reward
function hχ(y) is decreasing in y. To exclude the scenario where it is optimal
never to start, the condition stated in (2.1.10), namely, supy∈R+ hχ(y) > 0, is
now equivalent to
V χ(0) =bχ∗ − csF χ(bχ∗)
> cb, (2.2.22)
since F χ(0) = 1.
Lemma 2.2.10. For all y ≥ 0, the value function satisfies the inequality
0 ≤ Jχ(y) ≤ ( bχ∗−cFχ(bχ∗)
− cb)+.
Theorem 2.2.11. The optimal starting problem (2.1.8) admits the solution
Jχ(y) =
Vχ(y)− (y + cb) if y ∈ [0, dχ∗],
V χ(dχ∗)−(dχ∗+cb)Gχ(dχ∗)
Gχ(y) if y ∈ (dχ∗,+∞).
The optimal starting level dχ∗ > 0 is uniquely determined from
Gχ(d)(V χ′(d)− 1) = Gχ′(d)(V χ(d)− (d+ cb)).
As a result, it is optimal to start as soon as the CIR process Y falls below
the strictly positive level dχ∗.
2.2.2.2 Optimal Switching Problem
Now we study the optimal switching problem under the CIR model in (2.1.2).
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 30
Lemma 2.2.12. For all y ≥ 0, the value functions Jχ and V χ satisfy the
inequalities
0 ≤ Jχ(y) ≤ µθ
r,
0 ≤ V χ(y) ≤ y +2µθ
r.
We start by giving conditions under which it is optimal not to start ever.
Theorem 2.2.13. Under the CIR model, if it holds that
(i) yb ≤ 0, or
(ii) yb > 0 and cb ≥ bχ∗−csFχ(bχ∗)
,
with bχ∗ given in (2.2.21), then the optimal switching problem (2.1.13)-(2.1.14)
admits the solution
Jχ(y) = 0 for y ≥ 0, (2.2.23)
and
V χ(y) =
bχ∗−csFχ(bχ∗)
F χ(y) if y ∈ [0, bχ∗),
y − cs if y ∈ [bχ∗,+∞).
(2.2.24)
Conditions (i) and (ii) depend on problem data and can be easily verified.
In particular, recall that yb is defined in (2.2.20) and is easy to compute,
furthermore it is independent of σ and cs. Since it is optimal to never enter,
the switching problem is equivalent to a stopping problem and the solution in
Theorem 2.2.13 agrees with that in Theorem 2.2.9. Next, we provide conditions
under which it is optimal to enter as soon as the CIR process reaches some
lower level.
Theorem 2.2.14. Under the CIR model, if
yb > 0 and cb <bχ∗ − csF χ(bχ∗)
, (2.2.25)
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 31
with bχ∗ given in (2.2.21), then the optimal switching problem (2.1.13)-(2.1.14)
admits the solution
Jχ(y) =
PχF χ(y)− (y + cb) if y ∈ [0, dχ∗],
QχGχ(y) if y ∈ (dχ∗,+∞),
(2.2.26)
and
V χ(y) =
PχF χ(y) if y ∈ [0, bχ∗),
QχGχ(y) + (y − cs) if y ∈ [bχ∗,+∞),
(2.2.27)
where
P χ =Gχ(dχ∗)− (dχ∗ + cb)G
χ′(dχ∗)
F χ′(dχ∗)Gχ(dχ∗)− F χ(dχ∗)Gχ′(dχ∗),
Qχ =F χ(dχ∗)− (dχ∗ + cb)F
χ′(dχ∗)
F χ′(dχ∗)Gχ(dχ∗)− F χ(dχ∗)Gχ′(dχ∗).
There exist unique optimal starting and stopping levels dχ∗ and bχ∗, which are
found from the nonlinear system of equations:
Gχ(d)− (d+ cb)Gχ′(d)
F χ′(d)Gχ(d)− F χ(d)Gχ′(d)=
Gχ(b)− (b− cs)Gχ′(b)
F χ′(b)Gχ(b)− F χ(b)Gχ′(b),
F χ(d)− (d+ cb)Fχ′(d)
F χ′(d)Gχ(d)− F χ(d)Gχ′(d)=
F χ(b)− (b− cs)F χ′(b)
F χ′(b)Gχ(b)− F χ(b)Gχ′(b).
Moreover, we have that dχ∗ < yb and bχ∗ > ys.
In this case, it is optimal to start and stop an infinite number of times
where we start as soon as the CIR process drops to dχ∗ and stop when the
process reaches bχ∗. Note that in the case of Theorem 2.2.13 where it is never
optimal to start, the optimal stopping level bχ∗ is the same as that of the
optimal stopping problem in Theorem 2.2.9. The optimal starting level dχ∗,
which only arises when it is optimal to start and stop sequentially, is in general
not the same as dχ∗ in Theorem 2.2.11.
We conclude the section with two remarks.
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 32
Remark 2.2.15. Given the model parameters, in order to identify which of
Theorem 2.2.13 or Theorem 2.2.14 applies, we begin by checking whether yb ≤0. If so, it is optimal not to enter. Otherwise, Theorem 2.2.13 still applies
if cb ≥ bχ∗−csFχ(bχ∗)
holds. In the other remaining case, the problem is solved as in
Theorem 2.2.14. In fact, the condition cb <bχ∗−csFχ(bχ∗)
implies yb > 0 (see the
proof of Lemma 4.3 in Leung et al. [2014]. Therefore, condition (2.2.25) in
Theorem 2.2.14 is in fact identical to (2.2.22) in Theorem 2.2.11.
Remark 2.2.16. To verify the optimality of the results in Theorems 2.2.13
and 2.2.14, one can show by direct substitution that the solutions (Jχ, V χ) in
(2.2.23)-(2.2.24) and (2.2.26)-(2.2.27) satisfy the variational inequalities:
minrJχ(y)− LχJχ(y), Jχ(y)− (V χ(y)− (y + cb)) = 0,
minrV χ(y)− LχV χ(y), V χ(y)− (Jχ(y) + (y − cs)) = 0.
Indeed, this is the approach used by Zervos et al. [2013] for checking the solu-
tions of their optimal switching problems.
2.2.2.3 Numerical Examples
We numerically implement Theorems 2.2.9, 2.2.11, and 2.2.14, and illustrate
the associated starting and stopping thresholds. In Figure 2.4 (left), we ob-
serve the changes in optimal starting and stopping levels as speed of mean
reversion increases. Both starting levels dχ∗ and dχ∗ rise with µ, from 0.0964
to 0.1219 and from 0.1460 to 0.1696, respectively, as µ increases from 0.3 to
0.85. The optimal switching stopping level bχ∗ also increases. On the other
hand, stopping level bχ∗ for the starting-stopping problem stays relatively con-
stant as µ changes.
In Figure 2.4 (right), we see that as the stopping cost cs increases, the
increase in the optimal stopping levels is accompanied by a fall in optimal
starting levels. In particular, the stopping levels, bχ∗ and bχ∗ increase. In
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 33
comparison, both starting levels dχ∗ and dχ∗ fall. The lower starting level and
higher stopping level mean that the entry and exit times are both delayed as a
result of a higher transaction cost. Interestingly, although the cost cs applies
only when the process is stopped, it also has an impact on the timing to start,
as seen in the changes in dχ∗ and dχ∗ in the figure.
In Figure 2.4, we can see that the continuation (waiting) region of the
switching problem (dχ∗, bχ∗) lies within that of the starting-stopping problem
(dχ∗, bχ∗). The ability to enter and exit multiple times means it is possible to
earn a smaller reward on each individual start-stop sequence while maximizing
aggregate return. Moreover, we observe that optimal entry and exit levels of
the starting-stopping problem is less sensitive to changes in model parameters
than the entry and exit thresholds of the switching problem.
Figure 2.5 shows a simulated CIR path along with optimal entry and exit
levels for both starting-stopping and switching problems. Under the starting-
stopping problem, it is optimal to start once the process reaches dχ∗ = 0.0373
and to stop when the process hits bχ∗ = 0.4316. For the switching problem, it
is optimal to start once the process values hits dχ∗ = 0.1189 and to stop when
the value of the CIR process rises to bχ∗ = 0.2078. We note that both stopping
levels bχ∗ and bχ∗ are higher than the long-run mean θ = 0.2, and the starting
levels dχ∗ and dχ∗ are lower than θ. The process starts at Y0 = 0.15 > dχ∗,
under the optimal switching setting, the first time to enter occurs on day 8
when the process falls to 0.1172 and subsequently exits on day 935 at a level
of 0.2105. For the starting-stopping problem, entry takes place much later on
day 200 when the process hits 0.0306 and exits on day 2671 at 0.4369. Under
the optimal switching problem, two entries and two exits will be completed
by the time a single entry-exit sequence is realized for the starting-stopping
problem.
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 34
0.3 0.4 0.5 0.6 0.7 0.80.05
0.1
0.15
0.2
0.25
0.3
µ
dχ∗
bχ∗
dχ∗
bχ∗
0.01 0.02 0.03 0.04 0.050.1
0.15
0.2
0.25
cs
dχ∗
bχ∗
dχ∗
bχ∗
Figure 2.4: (Left) The optimal starting and stopping levels vs speed of mean rever-
sion µ. Parameters: σ = 0.15, θ = 0.2, r = 0.05, cs = 0.001, cb = 0.001. (Right) The
optimal starting and stopping levels vs transaction cost cs. Parameters: µ = 0.6,
σ = 0.15, θ = 0.2, r = 0.05, cb = 0.001.
2.3 Methods of Solution and Proofs
We now provide detailed proofs for our analytical results in Section 2.2 for
the optimal switching problems. Using the results for the double stopping
problems i.e. Theorems 2.2.2 and 2.2.4 under the XOU model and Theorems
2.2.9 and 2.2.11 under the CIR model, we can infer the structure of the buy and
sell regions of the switching problems and then proceed to verify its optimality.
2.3.1 Optimal Switching Timing under the XOU Model
In this section, we provide detailed proofs for Theorems 2.2.6 and 2.2.7.
Proof of Theorem 2.2.6 (Part 1) First, with hξs(x) = ex − cs, we differ-
entiate to get (hξsF
)′(x) =
(ex − cs)F ′(x)− exF (x)
F 2(x). (2.3.1)
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 35
0 1000 2000 3000 4000 50000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Days
bχ∗
b∗
d∗dχ∗
Figure 2.5: A sample CIR path, along with starting and stopping levels. Under
the starting-stopping setting, a starting decision is made at νdχ∗ = inft ≥ 0 : Yt ≤dχ∗ = 0.0373, and a stopping decision is made at τbχ∗ = inft ≥ νdχ∗ : Yt ≥bχ∗ = 0.4316. Under the optimal switching problem, entry and exit take place at
νdχ∗
= inft ≥ 0 : Yt ≤ dχ∗ = 0.1189 , and τbχ∗
= inft ≥ νdχ∗
: Yt ≥ bχ∗ = 0.2078respectively. Parameters: µ = 0.2, σ = 0.3, θ = 0.2, r = 0.05, cs = 0.001, cb = 0.001.
On the other hand, by Ito’s lemma, we have
hξs(x) = Exe−rthξs(Xt) − Ex∫ t
0
e−ru(L − r)hξs(Xu)du
.
Note that
Exe−rthξs(Xt) = e−rt(e(x−θ)e−µt+θ+σ2
4µ(1−e−2µt) − cs
)→ 0 as t→ +∞.
This implies that
hξs(x) = −Ex∫ +∞
0
e−ru(L − r)hξs(Xu)du
= −G(x)
∫ x
−∞Ψ(s)(L − r)hξs(s)ds
− F (x)
∫ +∞
x
Φ(s)(L − r)hξs(s)ds, (2.3.2)
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 36
where Ψ is defined in (2.2.16) and
Φ(x) :=2G(x)
σ2W(x).
The last line follows from Theorem 50.7 in Rogers and Williams [2000, p. 293].
Dividing both sides by F (x) and differentiating the RHS of (2.3.2), we obtain(hξsF
)′(x) = −
(G
F
)′(x)
∫ x
−∞Ψ(s)(L − r)hξs(s)ds
− G
F(x)Ψ(x)(L − r)hξs(x)− Φ(x)(L − r)hξs(x)
=W(x)
F 2(x)
∫ x
−∞Ψ(s)(L − r)hξs(s)ds =
W(x)
F 2(x)q(x),
where
q(x) :=
∫ x
−∞Ψ(s)(L − r)hξs(s)ds.
SinceW(x), F (x) > 0, we deduce that(hξsF
)′(x) = 0 is equivalent to q(x) = 0.
Using (2.3.1), we now see that (2.2.3) is equivalent to q(b) = 0.
Next, it follows from (2.2.6) that
q′(x) = Ψ(x)(L − r)hξs(x)
> 0 if x < xs,
< 0 if x > xs.
(2.3.3)
This, together with the fact that limx→−∞ q(x) = 0, implies that there exists
a unique bξ∗ such that q(bξ∗) = 0 if and only if limx→+∞ q(x) < 0. Next, we
show that this inequality holds. By the definition of hξs and F , we have
hξs(x)
F (x)=ex − csF (x)
> 0 for x > ln cs, limx→+∞
hξs(x)
F (x)= 0,(
hξsF
)′(x) =
W(x)
F 2(x)
∫ x
−∞Ψ(s)(L − r)hξs(s)ds =
W(x)
F 2(x)q(x). (2.3.4)
Since q is strictly decreasing in (xs,+∞), the above hold true if and only if
limx→+∞ q(x) < 0. Therefore, we conclude that there exits a unique bξ∗ such
that ebF (b) = (eb − cs)F ′(b). Using (2.3.3), we see that
bξ∗ > xs and q(x) > 0 for all x < bξ∗. (2.3.5)
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 37
Observing that ebξ∗, F (bξ∗), F ′(bξ∗) > 0, we can conclude that hξs(b
ξ∗) = ebξ∗ −
cs > 0, or equivalently bξ∗ > ln cs.
We now verify by direct substitution that V ξ(x) and Jξ(x) in (2.2.7) satisfy
the pair of variational inequalities:
minrJξ(x)− LJξ(x), Jξ(x)− (V ξ(x)− hξb(x)) = 0, (2.3.6)
minrV ξ(x)− LV ξ(x), V ξ(x)− (Jξ(x) + hξs(x)) = 0. (2.3.7)
First, note that Jξ(x) is identically 0 and thus satisfies the equality
(r − L)Jξ(x) = 0. (2.3.8)
To show that Jξ(x) − (V ξ(x) − hξb(x)) ≥ 0, we look at the disjoint intervals
(−∞, bξ∗) and [bξ∗,∞) separately. For x ≥ bξ∗, we have
V ξ(x)− hξb(x) = −(cb + cs),
which implies Jξ(x) − (V ξ(x) − hξb(x)) = cb + cs ≥ 0. When x < bξ∗, the
inequality
Jξ(x)− (V ξ(x)− hξb(x)) ≥ 0
can be rewritten as
hξb(x)
F (x)=ex + cbF (x)
≥ ebξ∗ − csF (bξ∗)
=hξs(b
ξ∗)
F (bξ∗). (2.3.9)
To determine the necessary conditions for this to hold, we consider the deriva-
tive of the LHS of (2.3.9):(hξbF
)′(x) =
W(x)
F 2(x)
∫ x
−∞Ψ(s)(L − r)hξb(s)ds (2.3.10)
=W(x)
F 2(x)
∫ x
−∞Ψ(s)esfb(s)ds.
If fb(x) = 0 has no roots, then (L − r)hξb(x) is negative for all x ∈ R. On the
other hand, if there is only one root x, then (L−r)hξb(x) = 0 and (L−r)hξb(x) <
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 38
0 for all other x. In either case, hξb(x)/F (x) is a strictly decreasing function
and (2.3.9) is true.
Otherwise if fb(x) = 0 has two distinct roots xb1 and xb2 with xb1 < xb2,
then
(L − r)hξb(x)
< 0 if x ∈ (−∞, xb1) ∪ (xb2,+∞),
> 0 if x ∈ (xb1, xb2).
(2.3.11)
Applying (2.3.11) to (2.3.10), the derivative (hξb/F )′(x) is negative on (−∞, xb1)
since the integrand in (2.3.10) is negative. Hence, hξb(x)/F (x) is strictly de-
creasing on (−∞, xb1). We further note that bξ∗>xs>xb2. Observe that on
the interval (xb1, xb2), the intergrand is positive. It is therefore possible for
(hξb/F )′ to change sign at some x ∈ (xb1, xb2). For this to happen, the positive
part of the integral must be larger than the absolute value of the negative part.
In other words, (2.2.15) must hold. If (2.2.15) holds, then there must exist
some a∗ ∈ (xb1, xb2) such that (hξb/F )′(a∗) = 0, or equivalently (2.2.8) holds:(hξbF
)′(a∗) =
hξ′
b (a∗)
F (a∗)− hξb(a
∗)F ′(a∗)
F 2(a∗)=
ea∗
F (a∗)− (ea
∗+ cb)F (a∗)′
F 2(a∗).
If (2.2.8) holds, then we have∣∣∣∣∫ xb1
−∞Ψ(x)exfb(x)dx
∣∣∣∣ =
∫ a∗
xb1
Ψ(x)exfb(x)dx.
In addition, since ∫ xb2
a∗Ψ(x)exfb(x)dx > 0,
it follows that ∣∣∣∣∫ xb1
−∞Ψ(x)exfb(x)dx
∣∣∣∣ < ∫ xb2
xb1
Ψ(x)exfb(x)dx.
This establishes the equivalence between (2.2.8) and (2.2.15). Under this con-
dition, hξb/F is strictly decreasing on (xb1, a∗). Then, either it is strictly in-
creasing on (a∗, bξ∗), or there exists some x ∈ (xb2, bξ∗) such that hξb(x)/F (x) is
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 39
strictly increasing on (a∗, x) and strictly decreasing on (x, bξ∗). In both cases,
(2.3.9) is true if and only if (2.2.9) holds.
Alternatively, if (2.2.15) doesn’t hold, then by (2.3.10), the integral (hξb/F )′
will always be negative. This means that the function hξb(x)/F (x) is strictly
decreasing for all x ∈ (−∞, bξ∗), in which case (2.3.9) holds.
We are thus able to show that (2.3.6) holds, in particular the minimum of
0 is achieved as a result of (2.3.8). To prove (2.3.7), we go through a similar
procedure. To check that
(r − L)V ξ(x) ≥ 0
holds, we consider two cases. First when x < bξ∗, we get
(r − L)V ξ(x) =ebξ∗ − csF (bξ∗)
(r − L)F (x) = 0.
On the other hand, when x ≥ bξ∗, the inequality holds
(r − L)V ξ(x) = (r − L)hξs(x) > 0,
since bξ∗ > xs (the first inequality of (2.3.5)) and (2.2.6).
Similarly, when x ≥ bξ∗, we have
V ξ(x)− (Jξ(x) + hξs(x)) = hξs(x)− hξs(x) = 0.
When x < bξ∗, the inequality holds:
V ξ(x)− (Jξ(x) + hξs(x)) =hξs(b
ξ∗)
F (bξ∗)F (x)− hξs(x) ≥ 0,
which is equivalent to hξs(x)F (x)≤ hξs(b
ξ∗)F (bξ∗)
, due to (2.3.4) and (2.3.5).
Proof of Theorem 2.2.7 (Part 1) Define the functions
qG(x, z) =
∫ +∞
x
Φ(s)(L − r)hξb(s)ds−∫ +∞
z
Φ(s)(L − r)hξs(s)ds,
qF (x, z) =
∫ x
−∞Ψ(s)(L − r)hξb(s)ds−
∫ z
−∞Ψ(s)(L − r)hξs(s)ds.
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 40
where We look for the points d∗ < b∗ such that
qG(d∗, b∗) = 0, and qF (d∗, b∗) = 0.
This is because these two equations are equivalent to (2.2.13) and (2.2.14),
respectively.
Now we start to solve the equations by first narrowing down the range for
d∗ and b∗. Observe that
qG(x, z) =
∫ z
x
Φ(s)(L − r)hξb(s)ds+
∫ ∞z
Φ(s)[(L − r)(hξb(s)− hξs(s)]ds
=
∫ z
x
Φ(s)(L − r)hξb(s)ds− r(cb + cs)
∫ ∞z
Φ(s)ds
< 0, (2.3.12)
for all x and z such that xb2 ≤ x < z. Therefore, d∗ ∈ (−∞, xb2).
Since bξ∗ > xs satisfies q(bξ∗) = 0 and a∗ < xb2 satisfies (2.2.8), we have
limz→+∞
qF (x, z) =
∫ x
−∞Ψ(s)(L − r)hξb(s)ds− q(bξ∗)−
∫ +∞
bξ∗Ψ(s)(L − r)hξs(s)ds
> 0,
for all x ∈ (a∗, xb2). Also, we note that
∂qF∂z
(x, z) = −Ψ(z)(L − r)hξs(z)
< 0 if z < xs,
> 0 if z > xs,
(2.3.13)
and
qF (x, x) =
∫ x
−∞Ψ(s)(L − r)
[hξb(s)− hξs(s)
]ds
= −r(cb + cs)
∫ x
−∞Ψ(s)ds < 0. (2.3.14)
Then, (2.3.13) and (2.3.14) imply that there exists a unique function β :
[a∗, xb2) 7→ R s.t. β(x) > xs and
qF (x, β(x)) = 0. (2.3.15)
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 41
Differentiating (2.3.15) with respect to x, we see that
β′(x) =Ψ(x)(L − r)hξb(x)
Ψ(β(x))(L − r)hξs(β(x))< 0,
for all x ∈ (xb1, xb2). In addition, by the facts that bξ∗ > xs satisfies q(bξ∗) = 0,
a∗ satisfies (2.2.8), and the definition of qF , we have
β(a∗) = bξ∗.
By (2.3.12), we have limx↑xb2 qG(x, β(x)) < 0. By computation, we get that
d
dxqG(x, β(x)) = −Φ(x)Ψ(β(x))− Φ(β(x))Ψ(x)
Ψ(β(x))(L − r)hξb(x)
= −Ψ(x)
[G(x)
F (x)− G(β(x))
F (β(x))
](L − r)hξb(x) < 0,
for all x ∈ (xb1, xb2). Therefore, there exists a unique d∗ such that qG(d∗, β(d∗)) =
0 if and only if
qG(a∗, β(a∗)) > 0.
The above inequality holds if (2.2.10) holds. Indeed, direct computation yields
the equivalence:
qG(a∗, β(a∗))
=
∫ +∞
a∗Φ(s)(L − r)hξb(s)ds−
∫ +∞
bξ∗Φ(s)(L − r)hξs(s)ds
= −hξb(a∗)
F (a∗)− G(bξ∗)
F (bξ∗)
∫ bξ∗
−∞Ψ(s)(L − r)hξs(s)ds−
∫ +∞
bξ∗Φ(s)(L − r)hξs(s)ds
= −ea∗ + cbF (a∗)
+ebξ∗ − csF (bξ∗)
.
When this solution exists, we have
d∗ ∈ (xb1, xb2) and b∗ := β(d∗) > xs.
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 42
Next, we show that the functions Jξ and V ξ given in (2.2.11) and (2.2.12)
satisfy the pair of VIs in (2.3.6) and (2.3.7). In the same vein as the proof for
Theorem 2.2.6, we show
(r − L)Jξ(x) ≥ 0
by examining the 3 disjoint regions on which Jξ(x) assume different forms.
When x < a∗,
(r − L)Jξ(x) = P (r − L)F (x) = 0.
Next, when x > d∗,
(r − L)Jξ(x) = Q(r − L)G(x) = 0.
Finally for x ∈ [a∗, d∗],
(r − L)Jξ(x) = (r − L)(KF (x)− hξb(x)) = −(r − L)hξb(x) > 0,
as a result of (2.3.11) since a∗, d∗ ∈ (xb1, xb2).
Next, we verify that
(r − L)V ξ(x) ≥ 0.
Indeed, we have (r − L)V ξ(x) = K(r − L)F (x) = 0 for x < b∗. When x ≥ b∗,
we get the inequality (r−L)V ξ(x) = (r−L)(QG(x)+hξs(x)) = (r−L)hξs(x) > 0
since b∗ > xs and due to (2.2.6).
It remains to show that Jξ(x)− (V ξ(x)− hξb(x)) ≥ 0 and V ξ(x)− (Jξ(x) +
hξs(x)) ≥ 0. When x < a∗, we have
Jξ(x)− (V ξ(x)− hξb(x)) = (P − K)F (x) + (ex + cb)
= −F (x)ea∗
+ cbF (a∗)
+ (ex + cb) ≥ 0.
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 43
This inequality holds since we have shown in the proof of Theorem 2.2.6 that
hξb(x)
F (x)is strictly decreasing for x < a∗. In addition,
V ξ(x)− (Jξ(x) + hξs(x)) = F (x)ea∗
+ cbF (a∗)
− (ex − cs) ≥ 0,
since (2.3.3) (along with the ensuing explanation) implies that hξs(x)F (x)
is increas-
ing for all x ≤ a∗.
In the other region where x ∈ [a∗, d∗], we have
Jξ(x)− (V ξ(x)− hξb(x)) = 0,
V ξ(x)− (Jξ(x) + hξs(x)) = hξb(x)− hξs(x) = cb + cs ≥ 0.
When x > b∗, it is clear that
Jξ(x)− (V ξ(x)− hξb(x)) = hξb(x)− hξs(x) = cb + cs ≥ 0,
V ξ(x)− (Jξ(x) + hξs(x)) = 0.
To establish the inequalities for x ∈ (d∗, b∗), we first denote
gJξ(x) := Jξ(x)− (V ξ(x)− hξb(x)) = QG(x)− KF (x) + hξb(x)
= F (x)
∫ x
d∗Φ(s)(L − r)hξb(s)ds−G(x)
∫ x
d∗Ψ(s)(L − r)hξb(s)ds,
gV ξ(x) := V ξ(x)− (Jξ(x) + hξs(x)) = KF (x)− QG(x)− hξs(x)
= F (x)
∫ b∗
x
Φ(s)(L − r)hξs(s)ds−G(x)
∫ b∗
x
Ψ(s)(L − r)hξs(s)ds.
In turn, we compute to get
g′Jξ
(x) = F ′(x)
∫ x
d∗Φ(s)(L − r)hξb(s)ds−G′(x)
∫ x
d∗Ψ(s)(L − r)hξb(s)ds,
g′V ξ
(x) = F ′(x)
∫ b∗
x
Φ(s)(L − r)hξs(s)ds−G′(x)
∫ b∗
x
Ψ(s)(L − r)hξs(s)ds.
Recall the definition of xb2 and xs, and the fact that G′ < 0 < F ′, we have
g′Jξ
(x) > 0 for x ∈ (d∗, xb2) and g′V ξ
(x) < 0 for x ∈ (xs, b∗). These, together
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 44
with the fact that gJξ(d∗) = gV ξ (b
∗) = 0, imply that
gJξ(x) > 0 for x ∈ (d∗, xb2), and gV ξ(x) > 0 for x ∈ (xs, b∗).
Furthermore, since we have
gJξ (b∗) = cb + cs ≥ 0, gV ξ(d
∗) = cb + cs ≥ 0, (2.3.16)
and
(L − r)gJξ(x) = (L − r)hξb(x) < 0 for all x ∈ (xb2, b∗),
(L − r)gV ξ(x) = −(L − r)hξs(x) < 0 for all x ∈ (d∗, xs). (2.3.17)
In view of inequalities (2.3.16)–(2.3.17), the maximum principle implies that
gJξ(x) ≥ 0 and gV ξ(x) ≥ 0 for all x ∈ (d∗, b∗). Hence, we conclude that
J(x)− (V (x)−hξb(x)) ≥ 0 and V (x)− (J(x) +hξs(x)) ≥ 0 hold for x ∈ (d∗, b∗).
Proof of Theorems 2.2.6 and 2.2.7 (Part 2) We now show that the
candidate solutions in Theorems 2.2.6 and 2.2.7, denoted by jξ and vξ, are
equal to the optimal switching value functions Jξ and V ξ in (2.1.11) and
(2.1.12), respectively. First, we note that jξ ≤ Jξ and vξ ≤ V ξ, since Jξ and
V ξ dominate the expected discounted cash low from any admissible strategy.
Next, we show the reverse inequaities. In Part 1, we have proved that
jξ and vξ satisfy the VIs (2.3.6) and (2.3.7). In particular, we know that
(r − L)jξ ≥ 0, and (r − L)vξ ≥ 0. Then by Dynkin’s formula and Fatou’s
lemma, as in Øksendal [2003, p. 226], for any stopping times ζ1 and ζ2 such
that 0 ≤ ζ1 ≤ ζ2 almost surely, we have the inequalities
Exe−rζ1 jξ(Xζ1) ≥ Exe−rζ2 jξ(Xζ2), (2.3.18)
Exe−rζ1 vξ(Xζ1) ≥ Exe−rζ2 vξ(Xζ2). (2.3.19)
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 45
For Λ0 = (ν1, τ1, ν2, τ2, . . . ), noting that ν1 ≤ τ1 almost surely, we have
jξ(x) ≥ Exe−rν1 jξ(Xν1) (2.3.20)
≥ Exe−rν1(vξ(Xν1)− hξb(Xν1)) (2.3.21)
≥ Exe−rτ1 vξ(Xτ1) − Exe−rν1hξb(Xν1) (2.3.22)
≥ Exe−rτ1(jξ(Xτ1) + hξs(Xτ1)) − Exe−rν1hξb(Xν1) (2.3.23)
= Exe−rτ1 jξ(Xτ1)+ Exe−rτ1hξs(Xτ1)− e−rν1hξb(Xν1), (2.3.24)
where (2.3.20) and (2.3.22) follow from (2.3.18) and (2.3.19) respectively. Also,
(2.3.21) and (2.3.23) follow from (2.3.6) and (2.3.7) respectively. Observe that
(2.3.24) is a recursion and jξ(x) ≥ 0 in both Theorems 2.2.6 and 2.2.7, we
obtain
jξ(x) ≥ Ex
∞∑n=1
[e−rτnhξs(Xτn)− e−rνnhξb(Xνn)]
.
Maximizing over all Λ0 yields that jξ(x) ≥ Jξ(x). A similar proof gives vξ(x) ≥V ξ(x).
Remark 2.3.1. If there is no transaction cost for entry, i.e. cb = 0, then
fb, which is now a linear function with a non-zero slope, has one root x0.
Moreover, we have fb(x) > 0 for x ∈ (−∞, x0) and fb(x) < 0 for x ∈ (x0,+∞).
This implies that the entry region must be of the form (−∞, d0), for some
number d0. Hence, the continuation region for entry is the connected interval
(d0,∞).
Remark 2.3.2. Let Lξ be the infinitesimal generator of the XOU process ξ =
eX , and define the function Hb(ς) := ς+cb ≡ hξb(ln ς). In other words, we have
the equivalence:
(Lξ − r)Hb(ς) ≡ (L − r)hξb(ln ς).
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 46
Referring to (2.2.5) and (2.2.6), we have either that
(Lξ − r)Hb(ς)
> 0 for ς ∈ (ςb1, ςb2),
< 0 for ς ∈ (0, ςb1) ∪ (ςb2,+∞),
(2.3.25)
where ςb1 = exb1 > 0 and ςb2 = exb2 and xb1 < xb2 are two distinct roots to
(2.2.5), or
(Lξ − r)Hb(ς) < 0, for ς ∈ (0, ς∗) ∪ (ς∗,+∞), (2.3.26)
where ς∗ = exb and xb is the single root to (2.2.5). In both cases, Assumption 4
of Zervos et al. [2013] is violated, and their results cannot be applied. Indeed,
they would require that (Lξ − r)Hb(ς) is strictly negative over a connected
interval of the form (ς0,∞), for some fixed ς0 ≥ 0. However, it is clear from
(2.3.25) and (2.3.26) that such a region is disconnected.
In fact, the approach by Zervos et al. [2013] applies to the optimal switching
problems where the optimal wait-for-entry region (in log-price) is of the form
(d∗,∞), rather than the disconnected region (−∞, a∗) ∪ (d∗,∞), as in our
case with an XOU underlying. Using the new inferred structure of the wait-
for-entry region, we have modified the arguments in Zervos et al. [2013] to
solve our optimal switching problem for Theorems 2.2.6 and 2.2.7.
2.3.2 Optimal Switching Timing under the CIR Model
Proofs of Theorems 2.2.13 and 2.2.14 Zervos et al. [2013] have studied
a similar problem of trading a mean-reverting asset with fixed transaction
costs, and provided detailed proofs using a variational inequalities approach.
In particular, we observe that yb and ys in (2.2.18) and (2.2.19) play the
same roles as xb and xs in Assumption 4 in Zervos et al. [2013], respectively.
However, Assumption 4 in Zervos et al. [2013] requires that 0 ≤ xb, this is
not necessarily true for yb in our problem. We have checked and realized that
CHAPTER 2. OPTIMAL SWITCHING UNDER XOU AND CIR 47
this assumption is not necessary for Theorem 2.2.13, and that yb < 0 simply
implies that there is no optimal starting level, i.e. it is never optimal to start.
In addition, Zervos et al. [2013] assume (in their Assumption 1) that the
hitting time of level 0 is infinite with probability 1. In comparison, we consider
not only the CIR case where 0 is inaccessible, but also when the CIR process
has a reflecting boundary at 0. In fact, we find that the proofs in Zervos
et al. [2013] apply to both cases under the CIR model. Therefore, apart from
relaxation of the aforementioned assumptions, the proofs of our Theorems
2.2.13 and 2.2.14 are the same as that of Lemmas 1 and 2 in Zervos et al.
[2013] respectively.
CHAPTER 3. SPECULATIVE FUTURES TRADING 48
Chapter 3
Speculative Futures Trading
under Mean Reversion
In this chapter, we study the problem of trading futures with transaction costs
when the underlying spot price is driven by an OU, CIR or XOU model. We
analytically derive the futures term structure and examine its connection to
futures price dynamics. For each futures contract, we describe the evolution of
the roll yield, and compute explicitly the expected roll yield. For the futures
trading problem, we first consider the standard entry and exit problems and
go on to incorporate a chooser option to either go long or short a futures
contract upon entry. We formulate the optimal double stopping problems
and solve them numerically using a finite difference method to determine the
optimal trading strategies. Numerical examples are provided to illustrate the
optimal entry and exit boundaries under different models and problem settings.
Our results show that the option to choose between a long or short position
motivates the investor to delay market entry, as compared to the case where
the investor pre-commits to go either long or short.
Section 3.1 summarizes the futures prices and term structures under mean
reversion. We discuss the concept of roll yield in Section 3.2. In Section 3.3,
CHAPTER 3. SPECULATIVE FUTURES TRADING 49
we formulate and numerically solve the optimal double stopping problems for
futures trading. The numerical algorithm is described in Appendix C.
3.1 Futures Prices and Term Structures
Throughout this chapter, we consider futures that are written on an asset
whose price process is mean-reverting. In this section, we discuss the pricing
of futures and their term structures under different spot models.
3.1.1 OU and CIR Spot Models
We begin with two mean-reverting models for the spot price S, namely, the
OU and CIR models. As we will see, they yield the same price function for
the futures contract. To start, suppose that the spot price evolves according
to the OU model:
dSt = µ(θ − St)dt+ σdBt,
where µ, σ > 0 are the speed of mean reversion and volatility of the process
respectively. θ ∈ R is the long run mean and B is a standard Brownian motion
under the historical measure P.
To price futures, we assume a re-parametrized OU model for the risk-
neutral spot price dynamics. Hence, under the risk-neutral measure Q, the
spot price follows
dSt = µ(θ − St) dt+ σ dBQt ,
with constant parameters µ, σ > 0, and θ ∈ R. This is again an OU process,
albeit with a different long-run mean θ and speed of mean reversion µ under
the risk-neutral measure. This involves a change of measure that connects the
CHAPTER 3. SPECULATIVE FUTURES TRADING 50
two Brownian motions, as described by
dBQt = dBt +
µ(θ − St)− µ(θ − St)σ
dt.
Throughout, futures prices are computed the same as forward prices, and
we do not distinguish between the two prices (see Cox et al. [1981]; Brennan
and Schwartz [1990]). As such, the price of a futures contract with maturity
T is given by
fTt ≡ f(t, St;T ) := EQST |St = (St − θ)e−µ(T−t) + θ, t ≤ T. (3.1.1)
Note that the futures price is a deterministic function of time and the current
spot price.
We now consider the CIR model for the spot price:
dSt = µ(θ − St)dt+ σ√StdBt, (3.1.2)
where µ, θ, σ > 0, and B is a standard Brownian motion under the historical
measure P. Under the risk-neutral measure Q,
dSt = µ(θ − St)dt+ σ√StdB
Qt , (3.1.3)
where µ, θ > 0, and BQ is a Q-standard Brownian motion. In both SDEs,
(3.1.2) and (3.1.3), we require 2µθ ≥ σ2 and 2µθ ≥ σ2 (Feller condition) so
that the CIR process stays positive.
The two Brownian motions are related by
dBQt = dBt +
µ(θ − St)− µ(θ − St)σ√St
dt,
which preserves the CIR model, up to different parameter values across two
measures.
The CIR terminal spot price ST admits the non-central Chi-squared dis-
tribution and is positive, whereas the OU spot price is normally distributed.
CHAPTER 3. SPECULATIVE FUTURES TRADING 51
Nevertheless, the futures price under the CIR model admits the same func-
tional form as in the OU case (see (3.1.1)):
fTt = (St − θ)e−µ(T−t) + θ, t ≤ T. (3.1.4)
Proposition 3.1.1. Under the OU or CIR spot model, the futures curve is (i)
upward-sloping and concave if the current spot price S0 < θ, (ii) downward-
slopping and convex if S0 > θ.
Proof. We differentiate (3.1.4) with respect to T to get the derivatives:
∂fT0∂T
= −µ(S0 − θ)e−µT ≶ 0 and∂2fT0∂T 2
= µ2(S0 − θ)e−µT ≷ 0,
for S0 ≷ θ. Hence, we conclude.
Remark 3.1.2. The futures price formula (3.1.4) holds more generally for
other mean-reverting models with risk-neutral spot dynamics of the form:
dSt = µ(θ − St)dt+ σ(St)dBQt ,
where σ(·) is a deterministic function such that EQ∫ T
0σ(St)
2dt <∞.
Under the OU model, the futures satisfies the following SDE under the
historical measure P:
dfTt =[(fTt − θ)(µ− µ) + µ(θ − θ)e−µ(T−t)
]dt+ σe−µ(T−t)dBt. (3.1.5)
If the spot follows a CIR process, then the futures prices follows
dfTt =[(fTt − θ)(µ− µ) + µ(θ − θ)e−µ(T−t)
]dt (3.1.6)
+ σe−µ(T−t)√
(fTt − θ)eµ(T−t) + θdBt.
Notice that the same drift appears in both (3.1.5) and (3.1.6). Alternatively,
we can express the drift in terms of the spot price as
e−µ(T−t)(µ(θ − St)− µ(θ − St)).
CHAPTER 3. SPECULATIVE FUTURES TRADING 52
This involves the difference between the mean-reverting drifts of the spot price
under the historical measure P and the risk-neutral measure Q. Therefore, the
drift of the futures price SDE is positive when the drift of the spot price under
P is greater than that under Q, i.e.
µ(θ − St) > µ(θ − St),
and vice versa.
Now, consider an investor with a long position in a single futures con-
tract, she wishes to close out the position and is interested in determining the
best time to short. We consider the delayed liquidation premium, which was
introduced in Leung and Shirai [2015] for equity options. This premium ex-
presses the benefit of waiting to liquidate as compared to closing the position
immediately. Precisely, the delayed liquidation premium is defined as
L(t, s) := supτ∈Tt,T
Et,se−r(τ−t)(f(τ, Sτ ;T )− c)
− (f(t, s;T )− c), (3.1.7)
where Tt,T is the set of all stopping times, with respect to the filtration gener-
ated by S, and c is the transaction cost. As we can see in (3.1.7), the optimal
stopping time for L(t, s), denoted by τ ∗, maximizes the expected discounted
value from liquidating the futures.
Proposition 3.1.3. Let t ∈ [0, T ] be the current time, and define the function
G(u, s) := e−µ(u−t)(µ(θ − s) + (r − µ)(θ − s)) + r(c− θ).
Under the OU spot model, if G(u, s) ≥ 0, ∀(u, s) ∈ [t, T ] × R, then it is
optimal to hold the futures contract till expiry, namely, τ ∗ = T in (3.1.7). If
G(u, s) < 0, ∀(u, s) ∈ [t, T ] × R, then it is optimal to liquidate immediately,
namely, τ ∗ = t. The same holds under the CIR model with G(u, s) defined over
[t, T ]× R+.
CHAPTER 3. SPECULATIVE FUTURES TRADING 53
Proof. Applying Ito’s formula to the process of e−rt(fTt − c) and taking expec-
tation, we can express (3.1.7) as
L(t, s) = supτ∈Tt,T
Et,s∫ τ
t
e−r(u−t)[e−µ(u−t)(µ(θ − Su) (3.1.8)
+ (r − µ)(θ − Su)) + r(c− θ)]du.
Therefore, if G(u, s) (the integrand in (3.1.8)) is positive, ∀(u, s) ∈ [t, T ]× R,
then the delayed liquidation premium can be maximized by choosing τ ∗ = T,
which is the largest stopping time. Conversely, if G < 0 ∀(u, s) ∈ [t, T ] × R,
then it is optimal to take τ ∗ = t in (3.1.8). Note that if G = 0 ∀(u, s) ∈[t, T ] × R, then the delayed liquidation premium is zero, and the investor is
indifferent toward when to liqudiate.
3.1.2 XOU Spot Model
Under the exponential OU (XOU) model, the spot price follows the SDE:
dSt = µ(θ − ln(St))Stdt+ σStdBt, (3.1.9)
with positive parameters (µ, θ, σ), and standard Brownian motion B under
the historical measure P. For pricing futures, we assume that the risk-neutral
dynamics of S satisfies
dSt = µ(θ − ln(St))Stdt+ σStdBQt ,
where µ, θ > 0, and BQ is a standard Brownian motion under the risk-neutral
measure Q.
For a futures contract written on S with maturity T , its price at time t is
given by
fTt = exp
(e−µ(T−t) ln(St) + (1− e−µ(T−t))(θ − σ2
2µ)
+σ2
4µ(1− e−2µ(T−t))
). (3.1.10)
CHAPTER 3. SPECULATIVE FUTURES TRADING 54
Consequently, the dynamics of the futures price under the historical measure
P is given as
dfTt =
[(ln(fTt ) + (e−µ(T−t) − 1)(θ − σ2
2µ) +
σ2
4µ(e−2µ(T−t) − 1)
)(µ− µ)
+ e−µ(T−t)(µθ − µθ)]fTt dt+ σe−µ(T−t)fTt dBt. (3.1.11)
By rearranging the first term in (3.1.11), the drift of the futures price SDE is
positive iff
fTt > exp
[e−µ(T−t)(µθ − µθ)
µ− µ − (e−µ(T−t) − 1)(θ − σ2
2µ)− σ2
4µ(e−2µ(T−t) − 1)
],
or equivalently in terms of the spot price,
St > exp
(µθ − µθµ− µ
). (3.1.12)
In particular, if θ = θ, condition (3.1.12) reduces to logSt > θ. Intuitively,
since the futures price must converge to the spot price at maturity, the futures
price tends to rise to approach the spot price when the spot price is high, as
observed in this condition.
We now consider the delayed liquidation premium defined in (3.1.7) but
under the XOU spot model. Applying Ito’s formula, we express the optimal
liquidation premium as
L(t, s) = supτ∈Tt,T
Et,s∫ τ
t
e−r(u−t)G(u, Su)du
, (3.1.13)
where
G(u, s) :=r +
[µ(θ − ln(s))− µ(θ − ln(s))
]e−µ(u−t)
× exp
(e−µ(u−t) ln(s) + (1− e−µ(u−t))(θ − σ2
2µ) +
σ2
4µ(1− e−2µ(u−t))
)− rc.
By inspecting the premium definition, we obtain the condition under which
immediate liquidation or waiting till maturity is optimal. The proof is identical
to that of Proposition 3.1.3, so we omit it.
CHAPTER 3. SPECULATIVE FUTURES TRADING 55
Proposition 3.1.4. Let t ∈ [0, T ] be the current time. Under the XOU spot
model, if G(u, s) ≥ 0 ∀(u, s) ∈ [t, T ]×R+, then holding till maturity (τ ∗ = T )
is optimal for (3.1.13). If G(u, s) < 0, ∀(u, s) ∈ [t, T ] × R+, then immediate
liquidation (τ ∗ = t) is optimal for (3.1.13).
Next, we summarize the term structure of futures under the XOU spot
model.
Proposition 3.1.5. Under the XOU spot model, the futures curve is
(i) downward-sloping and convex if
lnS0 > θ − σ2
2µ(1− e−µT ) +
(e2µT
4+σ2
2µ
) 12
− eµT
2,
(ii) downward-sloping and concave if
θ − σ2
2µ(1− e−µT ) < lnS0 < θ − σ2
2µ(1− e−µT ) +
(e2µT
4+σ2
2µ
) 12
− eµT
2,
(iii) upward-sloping and concave if
θ − σ2
2µ(1− e−µT )−
(e2µT
4+σ2
2µ
) 12
− eµT
2< lnS0 < θ − σ2
2µ(1− e−µT ),
and
(iv) upward-sloping and convex if
lnS0 < θ − σ2
2µ(1− e−µT )−
(e2µT
4+σ2
2µ
) 12
− eµT
2.
Proof. Direct differentiation of fT0 yields that
∂fT0∂T
=
[µ(θ − σ2
2µ− lnS0)e−µT +
σ2
2e−2µT
]fT0 ,
CHAPTER 3. SPECULATIVE FUTURES TRADING 56
and
∂2fT0∂T 2
=
[µ2e−2µT (θ − σ2
2µ− lnS0)2 + (µσ2e−3µT − µ2e−µT )(θ − σ2
2µ− lnS0)
+σ4
4e−4µT − σ2µe−2µT
]fT0 .
The results are obtained by analyzing the signs of the first and second order
derivatives.
CHAPTER 3. SPECULATIVE FUTURES TRADING 57
days to expiration0 50 100 150 200 250
VIX
futu
res
pric
e
40
45
50
55
60
65
70
75
80
85
CIR/OUXOUobserved price
days to expiration0 50 100 150 200 250
VIX
futu
res
pric
e
12
13
14
15
16
17
18
19
CIR/OUXOUobserved price
Figure 3.1: (Left) VIX futures historical prices on Nov 20, 2008 with the current
VIX value at 80.86. The days to expiration range from 26 to 243 days (Dec–Jul
contracts). Calibrated parameters: µ = 4.59, θ = 40.36 under the CIR/OU model,
or µ = 3.25, θ = 3.65, σ = 0.15 under the XOU model. (Right) VIX futures historical
prices on Jul 22, 2015 with the current VIX value at 12.12. The days to expiration
ranges from 27 days to 237 days (Aug–Mar contracts). Calibrated parameters:
µ = 4.55, θ = 18.16 under the CIR/OU model, or µ = 4.08, θ = 3.06, σ = 1.63 under
the XOU model.
Figure 3.1 displays two characteristically different term structures observed
in the VIX futures market. These futures, written on the CBOE Volatility
Index (VIX) are traded on the CBOE Futures Exchange. As the VIX measures
the 1-month implied volatility calculated from the prices of S&P 500 options,
VIX futures provide exposure to the market’s volatility. We plot the VIX
futures prices during the recent financial crisis on November 20, 2008 (left),
and on a post-crisis date, July 22, 2015 (right), along with the calibrated
futures curves under the OU/CIR model and XOU model. In the calibration,
the model parameter values are chosen to minimize the sum of squared errors
between the model and observed futures prices.
The OU/CIR/XOU model generates a decreasing convex curve for Novem-
ber 20, 2008 (left), and an increasing concave curve for July 22, 2015 (right),
CHAPTER 3. SPECULATIVE FUTURES TRADING 58
and they all fit the observed futures prices very well. The former term struc-
ture starts with a very high spot price of 80.86 with a calibrated risk-neutral
long-run mean θ = 40.36 under the OU/CIR model, suggesting that the mar-
ket’s expectation of falling market volatility. In contrast, we infer from the
term structure on July 25, 2015 that the market expects the VIX to raise from
the current spot value of 12.12 to be closer to θ = 18.16.
3.2 Roll Yield
By design, the value of a futures contract converges to the spot price as time
approaches maturity. If the futures market is in backwardation, the futures
price increases to reach the spot price at expiry. In contrast, when the market
is in contango, the futures price tends to decrease to the spot price. For an
investor with a long futures position, the return is positive in a backwardation
market, and negative in a contango market. An investor can long the front-
month contract, then short it at or before expiry, and simultaneously go long
the next-month contract. This rolling strategy that involves repeatedly rolling
an expiring contract into a new one is commonly adopted during backwarda-
tion, while its opposite is often used in a contango market. Backwardation
and contango phenomena are widely observed in the energy commodities and
volatility futures markets.
More generally, both the futures and spot prices vary over time. If the
spot price increases/decreases, the futures price will also end up higher/lower.
This leads us to consider the difference between the futures and spot returns,
defined as the change in values without dividing by the initial value.1 Let
0 ≤ t1 < t2 ≤ T . We denote the roll yield over the period [t1, t2] associated
1See Deconstructing Futures Returns: The Role of Roll Yield, Campbell White Paper
Series, February 2014.
CHAPTER 3. SPECULATIVE FUTURES TRADING 59
with a single futures contract with maturity T by
R(t1, t2, T ) := (fTt2 − fTt1)− (St2 − St1).
In other words, roll yield here is the change in the futures price that is not
accounted for by the change in spot price. It represents the net benefits and/or
costs of owning futures rather than the underlying asset itself.
This notion of roll yield is the same as that in Moskowitz et al. [2012] where
the relationship between roll yield and futures returns is studied. Gorton et al.
[2013] treat roll yield as the same as futures basis, which means a negative roll
yield signifies a market in contango and a positive roll yield is equivalent to
backwardation. In our set-up, if one always hold a futures contract to maturity,
then roll yield is the same as futures basis. Therefore, the definition of roll
yield in Gorton et al. [2013] is a special case of ours. In particular, if t2 = T ,
then the roll yield reduces to the price difference (St1 − fTt1). Furthermore,
observe that if St2 = St1 then roll yield becomes merely the change in futures
price.
A closely related concept is the S&P-GSCI roll yield. S&P-GSCI carries
out rolling of the underlying futures contracts once each month, from the fifth
to the ninth business day. On each day, 20% of the current portfolio is rolled
over, in a process commonly known as the Goldman roll. The S&P-GSCI
roll yield for each commodity is defined as the difference between the average
purchasing price of the new futures contracts and the average selling price of
the old futures contracts. In essence, it is an indicator of the sign of the slope
of the futures term structure. In comparison to the S&P-GSCI index, our
definition accounts for the changes of spot price over time.
Next, we examine the cumulative roll yield across maturities. Denote by
T1 < T2 < T3 < . . . the maturities of futures contracts. We roll over at every
Ti by replacing the contract expiring at Ti with a new contract that expires at
Ti+1. Let i(t) := mini : Ti−1 < t ≤ Ti, and i(0) = 1. Then the roll yield up
CHAPTER 3. SPECULATIVE FUTURES TRADING 60
to time t > T1 is
R(0, t) = (fTi(t)t − fTi(t)Ti(t)−1
) +
i(t)−1∑j=2
(STj − fTjTj−1
) + (ST1 − fT10 )− (St − S0)
= (fTi(t)t − St)− (fT10 − S0)︸ ︷︷ ︸
Basis Return
+
i(t)−1∑j=1
(STj − fTj+1
Tj)︸ ︷︷ ︸
Cumulative Roll Adjustment
. (3.2.1)
The cumulative roll adjustment is related to the term structure of futures
contracts. If Ti−Ti−1 is constant, and the term structure only moves parallel,
then the cumulative roll adjustment is simply the number of roll-over times a
constant (difference between spot and near-month futures contract).
3.2.1 OU and CIR Spot Models
Suppose the spot price follows the OU or CIR model described in Section 3.1.1.
Inspecting (3.2.1), we can write down the SDE for the roll yield under the OU
model:
dR(0, t) = dfTi(t)t − dSt
=[e−µ(Ti(t)−t)
(µ(θ − St)− µ(θ − St)
)− µ(θ − St)
]dt
+ σ(e−µ(Ti(t)−t) − 1
)dBt. (3.2.2)
The roll yield SDE under the CIR model has the same drift as (3.2.2). Fur-
thermore, the drift is positive iff
St >e−µ(Ti(t)−t)(µθ − µθ) + µθ
e−µ(Ti(t)−t)(µ− µ) + µ.
In particular, if θ = θ, then the drift is positive iff St > θ. When t = Ti(t), the
drift is µ(St − θ) and is positive iff St > θ. Furthermore, the drift term can
also be expressed as
µ(fTi(t)t − θ
)−(1− e−µ(Ti(t)−t)
)µ(θ − St).
CHAPTER 3. SPECULATIVE FUTURES TRADING 61
On the other hand, we observe that
dR(0, t)dSt = σ2(e−µ(Ti(t)−t) − 1
)dt,
under the OU case and
dR(0, t)dSt = σ2(e−µ(Ti(t)−t) − 1
)Stdt,
under the CIR case. In other words, the instantaneous covariations betweeen
roll yield and spot price under both OU and CIR models are negative for
t < Ti(t) regardless of the spot price level.
Consider a longer horizon with rolling at multiple maturities, the expected
roll yield is
ER(0, t) = EfTi(t)t − St − (fT10 − S0) +
i(t)−1∑j=1
ESTj − fTj+1
Tj
= ((S0 − θ)e−µt + θ − θ)(e−µ(Ti(t)−t) − 1)− (S0 − θ)(e−µT1 − 1)
+
i(t)−1∑j=1
((S0 − θ)e−µTj + θ − θ)(1− e−µ(Tj+1−Tj)).
In summary, the expected roll yield depends not only on the risk-neutral
parameters µ and θ, but also their historical counterparts. It vanishes when
S0 = θ = θ. This is intuitive because if the current spot price is currently at
the long-run mean, and the risk-neutral and historical measures coincide, then
the spot and futures prices have little tendency to deviate from the long-run
mean. Also, notice that neither the futures price nor the roll yield depends
on the volatility parameter σ. This is true under the OU/CIR model, but not
the exponential OU model, as we discuss next.
3.2.2 XOU Spot Model
We now turn to the exponential OU spot price model discussed in Section
3.1.2. Recalling the futures price in (3.1.10), the expected roll yield is given
CHAPTER 3. SPECULATIVE FUTURES TRADING 62
by
ER(0, t) = Y1(t) + Y2(t)− (fT10 − S0), (3.2.3)
where
Y1(t) = EfTi(t)t − St
= exp
(e−µ(Ti(t)−t)−µt ln(S0) +
(θ − σ2
2µ
)(1− e−µt)e−µ(Ti(t)−t)
+σ2
4µe−2µ(Ti(t)−t)(1− e−2µt) + (1− e−µ(Ti(t)−t))(θ − σ2
2µ)
+σ2
4µ(1− e−2µ(Ti(t)−t))
)− exp
(e−µt ln(S0) + (1− e−µt)(θ − σ2
2µ) +
σ2
4µ(1− e−µt)
),
and
Y2(t) =
i(t)−1∑j=1
ESTj − fTj+1
Tj
=
i(t)−1∑j=1
(exp
(e−µTj ln(S0) + (1− e−µTj)(θ − σ2
2µ) +
σ2
4µ(1− e−µTj)
)− exp
(e−µ(Tj+1−Tj)−µTj ln(S0) +
(θ − σ2
2µ
)(1− e−µTj)e−µ(Tj+1−Tj)
+σ2
4µe−2µ(Tj+1−Tj)(1− e−2µTj)
+ (1− e−µ(Tj+1−Tj))(θ − σ2
2µ) +
σ2
4µ(1− e−2µ(Tj+1−Tj))
)).
The explicit formula (3.2.3) for the expected roll yield reveals the non-
trivial dependence on the volatility parameter σ, as well as the risk-neutral
parameters (µ, θ) and historical parameters (µ, θ). It is useful for instantly
predicting the roll yield after calibrating the risk-neutral parameters from the
term structure of the futures prices, and estimating the historical parameters
from past spot prices.
CHAPTER 3. SPECULATIVE FUTURES TRADING 63
Referring to (3.1.9) and (3.1.11), the historical dynamics of the roll yield
under an XOU spot model is given by
dR(0, t) = h(t, s)dt+ σ(e−µ(Ti(t)−t)f
Ti(t)t − St
)dBt,
where
h(t, s) =(
ln s(µ− µ) + (µθ − µθ))
exp
(e−µ(Ti(t)−t) ln(s)
+ (1− e−µ(Ti(t)−t))(θ − σ2
2µ) +
σ2
4µ(1− e−2µ(Ti(t)−t))
)e−µ(Ti(t)−t)
− µ(θ − ln(s))s,
is the drift expressed in terms of the spot price St. This reduces to
h(Ti(t), s) = s ln(s)(µ− µ+ µθ)− µθs, if t = Ti(t).
Unlike the OU/CIR case, under an XOU spot model there is no explicit solu-
tion for the critical level of the spot price at which the drift changes sign.
As in the OU/CIR spot model, it is of interest to compute
dR(0, t)dSt = σ2(e−µ(Ti(t)−t)f
Ti(t)t − St
)Stdt,
from which we see that the covariation between roll yield and spot price can be
either positive or negative. In particular when the futures price is significantly
higher than the spot price, i.e. when the market is in contango, the correlation
tends to be positive.
3.3 Optimal Timing to Trade Futures
In Section 3.1, we have discussed the timing to liquidate a long futures position,
and the concept of rolling discussed in Section 3.2 corresponds to holding the
futures up to expiry. In this section, we further explore the timing options
embedded in futures, and develop the optimal trading strategies.
CHAPTER 3. SPECULATIVE FUTURES TRADING 64
3.3.1 Optimal Double Stopping Approach
Let us consider the scenario in which an investor has a long position in a
futures contract with expiration date T . With a long position in the futures,
the investor can hold it till maturity, but can also close the position early by
taking an opposite position at the prevailing market price. At maturity, the
two opposite positions cancel each other. This motivates us to investigate the
best time to close.
If the investor selects to close the long position at time τ ≤ T , then she
will receive the market value of the futures on the expiry date, denoted by
f(τ, Sτ ;T ), minus the transaction cost c ≥ 0. To maximize the expected
discounted value, evaluated under the investor’s historical probability measure
P with a constant subjective discount rate r > 0, the investor solves the
optimal stopping problem
V(t, s) = supτ∈Tt,T
Et,se−r(τ−t)(f(τ, Sτ ;T )− c)
,
where Tt,T is the set of all stopping times, with respect to the filtration gen-
erated by S, taking values between t and T , where T ∈ (0, T ] is the trading
deadline, which can equal but not exceed the futures’ maturity. Throughout
this chapter, we continue to use the shorthand notation Et,s· ≡ E·|St = sto indicate the expectation taken under the historical probability measure P.
The value function V(t, s) represents the expected liquidation value asso-
ciated with the long futures position. Prior to taking the long position in
futures, the investor, with zero position, can select the optimal timing to start
the trade, or not to enter at all. This leads us to analyze the timing option
inherent in the trading problem. Precisely, at time t ≤ T , the investor faces
the optimal entry timing problem
J (t, s) = supν∈Tt,T
Et,se−r(ν−t)(V(ν, Sν)− (f(ν, Sν ;T ) + c))+
,
CHAPTER 3. SPECULATIVE FUTURES TRADING 65
where c ≥ 0 is the transaction cost, which may differ from c. In other words,
the investor seeks to maximize the expected difference between the value func-
tion V(ν, Sν) associated with the long position and the prevailing futures price
f(ν, Sν ;T ). The value function J (t, s) represents the maximum expected value
of the trading opportunity embedded in the futures. We refer this “long to
open, short to close” strategy as the long-short strategy.
Alternatively, an investor may well choose to short a futures contract with
the speculation that the futures price will fall, and then close it out later by
establishing a long position.2 Given an investor who has a unit short position
in the futures contract, the objective is to minimize the expected discounted
cost to close out this position at/before maturity. The optimal timing strategy
is determined from
U(t, s) = infτ∈Tt,T
Et,se−r(τ−t)(f(τ, Sτ ;T ) + c)
.
If the investor begins with a zero position, then she can decide when to enter
the market by solving
K(t, s) = supν∈Tt,T
Et,se−r(ν−t)((f(ν, Sν ;T )− c)− U(ν, Sν))
+.
We call this “short to open, long to close” strategy as the short-long strategy.
When an investor contemplates entering the market, she can either long
or short first. Therefore, on top of the timing option, the investor has an
additional choice between the long-short and short-long strategies. Hence, the
investor solves the market entry timing problem:
P(t, s) = supς∈Tt,T
Et,se−r(ς−t)maxA(ς, Sς),B(ς, Sς)
, (3.3.1)
2By taking a short futures position, the investor is required to sell the underlying spot
at maturity at a pre-specified price. In contrast to the short sale of a stock, a short futures
does not involve share borrowing or re-purchasing.
CHAPTER 3. SPECULATIVE FUTURES TRADING 66
with two alternative rewards upon entry defined by
A(ς, Sς) := (V(ς, Sς)− (f(ς, Sς ;T ) + c))+ (long-short),
B(ς, Sς) := ((f(ς, Sς ;T )− c)− U(ς, Sς))+ (short-long).
3.3.2 Variational Inequalities & Optimal Trading Strate-
gies
In order to solve for the optimal trading strategies, we study the variational
inequalities corresponding to the value functions J , V , U , K and P . To this
end, we first define the operators:
L(1)· := −r ·+∂·∂t
+ µ(θ − s) ∂·∂s
+σ2
2
∂2·∂s2
, (3.3.2)
L(2)· := −r ·+∂·∂t
+ µ(θ − s) ∂·∂s
+σ2s
2
∂2·∂s2
,
L(3)· := −r ·+∂·∂t
+ µ(θ − ln s)∂·∂s
+σ2s2
2
∂2·∂s2
, (3.3.3)
corresponding to, respectively, the OU, CIR, and XOU models.
The optimal exit and entry problems J and V associated with the long-
short strategy are solved from the following pair of variational inequalities:
maxL(i)V(t, s) , (f(t, s;T )− c)− V(t, s)
= 0, (3.3.4)
maxL(i)J (t, s) , (V(t, s)− (f(t, s;T ) + c))+ − J (t, s)
= 0, (3.3.5)
for (t, s) ∈ [0, T ] × R, with i ∈ 1, 2, 3 representing the OU, CIR, or XOU
model respectively.3 Similarly, the reverse short-long strategy can be deter-
mined by numerically solving the variational inequalities satisfied by U and
3The spot price is positive, thus s ∈ R+, under the CIR and XOU models.
CHAPTER 3. SPECULATIVE FUTURES TRADING 67
K:
minL(i)U(t, s) , (f(t, s;T ) + c)− U(t, s)
= 0, (3.3.6)
maxL(i)K(t, s) , ((f(t, s;T )− c)− U(t, s))+ −K(t, s)
= 0. (3.3.7)
As V , J , U , and K are numerically solved, they become the input to the final
problem represented by the value function P . To determine the optimal timing
to enter the futures market, we solve the variational inequality
maxL(i)P(t, s) , maxA(t, s),B(t, s) − P(t, s)
= 0. (3.3.8)
The optimal timing strategies are described by a series of boundaries rep-
resenting the time-varying critical spot price at which the investor should
establish a long/short futures position. In the “long to open, short to close”
trading problem, where the investor pre-commits to taking a long position
first, the market entry timing is described by the “J ” boundary in Figure
3.2(a). The subsequent timing to exit the market is represented by the “V”
boundary in Figure 3.2(a). As we can see, the investor will long the futures
when the spot price is low, and short to close the position when the spot price
is high, confirming the buy-low-sell-high intuition.
If the investor adopts the short-long strategy, by which she will first short
a futures and subsequently close out with a long position, then the optimal
market entry and exit timing strategies are represented, respectively, by the
“K” and “U” boundaries in Figure 3.2(c). The investor will enter the market
by shorting a futures when the spot price is sufficiently high (at the “K”
boundary), and close it out when the spot price is low. Thus, the boundaries
reflect a sell-high-buy-low strategy.
When there are no transaction costs (see Figure 3.2(b) and 3.2(d)), the
waiting region shrinks for both strategies. Practically, this means that the
investor tends to enter and exit the market earlier, resulting in more rapid
CHAPTER 3. SPECULATIVE FUTURES TRADING 68
trades. This is intuitive as transaction costs discourage trades, especially near
expiry.
In the market entry problem represented by P(t, s) in (3.3.1), the investor
decides at what spot price to open a position. The corresponding timing strat-
egy is illustrated by two boundaries in Figure 3.2(e). The boundary labeled
as “P = A” (resp. “P = B”) indicates the critical spot price (as a function of
time) at which the investor enters the market by taking a long (resp. short)
futures position. The area above the “P = B” boundary is the “short-first”
region, whereas the area below the “P = A” boundary is the “long-first” re-
gion. The area between the two boundaries is the region where the investor
should wait to enter. The ordering of the regions is intuitive – the investor
should long the futures when the spot price is currently low and short it when
the spot price is high. As time approaches maturity, the value of entering the
market diminishes. The investor will not start a long/short position unless the
spot is very low/high close to maturity. Therefore, the waiting region expands
significantly near expiry.
CHAPTER 3. SPECULATIVE FUTURES TRADING 69
time (days)0 5 10 15 20
spot
pric
e
5
10
15
20
25
V
J
Wait
Long
Short
(a)
time (days)0 5 10 15 20
spot
pric
e
5
10
15
20
25
V
J
Wait
Long
Short
(b)
time (days)0 5 10 15 20
spot
pric
e
5
10
15
20
25
K
U
Wait
Long
Short
(c)
time (days)0 5 10 15 20
spot
pric
e
5
10
15
20
25
K
U
Short
Wait
Long
(d)
time (days)0 5 10 15 20
spot
pric
e
5
10
15
20
25
Short First
Wait
Long FirstP = A
P = B
(e)
time (days)0 5 10 15 20
spot
pric
e
5
10
15
20
25
Short First
Long First
P = B
P = A
Wait
(f)
Figure 3.2: Optimal long-short boundaries with/without transaction costs for fu-
tures trading under the CIR model in (a) and (b) respectively, optimal short-long
boundaries with/without transaction costs in (c) and (d) respectively, and optimal
boundaries with/without transaction costs in (e) and (f) respectively. Parameters:
T = 22252 , T = 66
252 , r = 0.05, σ = 5.33, θ = 17.58, θ = 18.16, µ = 8.57, µ = 4.55, c =
c = 0.005.
CHAPTER 3. SPECULATIVE FUTURES TRADING 70
time (days)0 5 10 15 20
spot
pric
e
5
10
15
20
25
P = B
V
K
J U
P = A
(a)
time (days)0 5 10 15 20
spot
pric
e
5
10
15
20
25
VK
J
P = A
P = B
U
(b)
Figure 3.3: Optimal boundaries with and without transaction costs for futures
trading under the CIR model in (a) and (b) respectively. Parameters: T = 22252 ,
T = 66252 , r = 0.05, σ = 5.33, θ = 17.58, θ = 18.16, µ = 8.57, µ = 4.55, c = c = 0.005.
CHAPTER 3. SPECULATIVE FUTURES TRADING 71
time (days)0 5 10 15 20
spot
pric
e
6
8
10
12
14
16
18
20
22
24
26
K
V
P = B
U
P = A
J
(a)
time (days)0 5 10 15 20
spot
pric
e
0
5
10
15
20
25
30
35
P = BK
P = AJ
U
V
(b)
Figure 3.4: Optimal boundaries with transaction costs for futures trading. (a) OU
spot model with σ = 18.7, θ = 17.58, θ = 18.16, µ = 8.57, µ = 4.55. (b) XOU spot
model with σ = 1.63, θ = 3.03, θ = 3.06, µ = 8.57, µ = 4.08. Common parameters:
T = 22252 , T = 66
252 , c = c = 0.005.
CHAPTER 3. SPECULATIVE FUTURES TRADING 72
spot price6 8 10 12 14 16 18 20 22 24
0
0.5
1
1.5
2
2.5
P
A
B
Figure 3.5: The value functions P, A, and B plotted against the spot price at time
0. The parameters are the same as those in Figure 3.4.
The investor’s exit strategy depends on the initial entry position. If the
investor enters by taking a long position (at the “P = A” boundary), then the
optimal exit timing to close her position is represented by the upper boundary
with label “V” in Figure 3.2(a). If the investor’s initial position is short, then
the optimal time to close by going long the futures is described by the lower
boundary with label “U” in Figure 3.2(c).
Since the value function P dominates both J and K due to the additional
flexibility, it is not surprising that the “P = A” boundary is lower than the
“J ” boundary, and the “P = B” boundary is higher than the “K” boundary,
as seen in Figure 3.3(a). This means that the embedded timing option to
choose between the two strategies (“long to open, short to close” or “short to
open, long to close”) induces the investor to delay market entry to wait for
better prices. This phenomenon is also observed for both OU and XOU spot
models in Figure 3.4. Figure 3.5 shows that the value function P dominates
B and A for all values of spot price. We can also see the regions where the
“P = A” (when the spot price is low) and “P = B” (when the spot price is
CHAPTER 3. SPECULATIVE FUTURES TRADING 73
high).
We see that Figure 3.4(b) is similar to Figure 3.3(a), in both CIR and XOU
cases, the difference between “U” boundary and the “J ” boundary is much
larger than the difference between the “V” boundary and the “K” boundary.
This means that the decision to choose either long-short or short-long has
a larger impact on the optimal price level to long futures compared to the
optimal level to short. On the other hand, in Figure 3.4(a), we observe a more
symmetric relationship between the long-short and short-long optimal exercise
boundaries. In particular, choosing one strategy or the other does not affect
the optimal price levels as much as CIR and XOU cases.
3.4 Conclusion
We have studied an optimal double stopping approach for trading futures
under a number of mean-reverting spot models. Our model yields trading
decisions that are consistent with the spot price dynamics and futures term
structure. Accounting for the timing options as well as the option to choose
between a long or short position, we find that it is optimal to delay market
entry, as compared to the case of committing to either go long or short a priori.
A natural direction for future research is to investigate the trading strate-
gies under a multi-factor or time-varying mean-reverting spot price model. To
this end, we include here some references that discuss the pricing aspect of
futures under such models, for example, Detemple and Osakwe [2000]; Lu and
Zhu [2009]; Zhu and Lian [2012]; Mencıa and Sentana [2013] for VIX futures,
Schwartz [1997]; Ribeiro and Hodges [2004] for commodities, and Monoyios
and Sarno [2002] for equity index futures. It is also of practical interest to
develop similar optimal multiple stopping approaches to trading commodi-
ties under mean-reverting spot models (Leung et al. [2015, 2014]), and credit
CHAPTER 3. SPECULATIVE FUTURES TRADING 74
derivatives trading (Leung and Liu [2012]).
CHAPTER 4. RISK AVERSE TIMING 75
Chapter 4
Optimal Risk Averse Timing of
an Asset Sale
In this chapter, we incorporate risk preference into optimal stopping problems.
We consider a risk-averse investor who has an exponential, power, or log utility.
We analyze the optimal timing to sell an asset when prices are driven by a
GBM or XOU process – to account for, respectively, the trending and mean-
reverting price dynamics. This gives rise to a total of six different scenarios.
In all cases, we derive the optimal thresholds to liquidate the asset and the
certainty equivalents associated with the timing option to sell. We compare
these results across models and utilities, with emphasis on their dependence
on asset price, risk aversion, and quantity. We find that the timing option
may render the investor’s value function and certainty equivalent non-concave
in price. Numerical results are presented to illustrate the investor’s strategies
and the premium associated with optimally timing to sell.
In Section 4.1, the asset sale problems are formulated for different utilities
and price dynamics. In Section 4.2, we present the solutions of the problems
and discuss the optimal selling strategies. We analyze the certainty equivalents
in Section 4.3. All proofs are included in Section 4.4.
CHAPTER 4. RISK AVERSE TIMING 76
4.1 Problem Overview
We consider a risk-averse asset holder (investor) with a subjective probability
measure P. For our optimal asset sale problems, we will study two models for
the risky asset price, namely, the geometric Brownian motion (GBM) model
and the exponential Ornstein-Uhlenbeck (XOU) model. First, the GBM price
process S satisfies
dSt = µSt dt+ σSt dBt,
with constant parameters µ ∈ R and σ > 0, where (Bt)t≥0 is a standard
Brownian motion under P. Under the second model, the XOU price process ξ
is defined by
ξt = eXt ,
dXt = κ(θ −Xt) dt+ η dBt, (4.1.1)
where the log-price X is an OU process with constant parameters κ, η > 0,
θ ∈ R.
The investor’s risk preference is modeled by three utility functions:
(1) Exponential utility
Ue(w) = 1− e−γw, for w ∈ R,
with the risk aversion parameter γ > 0;
(2) Log utility
Ul(w) = log(w), for w > 0;
(3) Power utility
Up(w) =wp
p, for w ≥ 0,
where p := 1−%, with the risk aversion parameter % ∈ [0, 1). In particular,
when p = 1, the power utility is linear, corresponding to zero risk aversion.
CHAPTER 4. RISK AVERSE TIMING 77
Denote by F the filtration generated by the Brownian motion B, and Tthe set of all F-stopping times. The investor seeks to maximize the expected
discounted utility from asset sale by selecting the optimal stopping time. De-
note by ν > 0 the quantity of the risky asset to be sold. For simplicity, we
limit our analysis to simultaneous liquidation of all ν units. The investor will
receive the utility value of Ui(νSτ ) or Ui(νξτ ), i ∈ e, l, p, under the GBM
and XOU model respectively, when all units are sold at time τ .
Therefore, the investor solves the optimal stopping problems under two
different price dynamics:
(GBM) Vi(s, ν) = supτ∈T
Ese−rτUi(νSτ )
, (4.1.2)
(XOU) Vi(z, ν) = supτ∈T
Eze−rτUi(νξτ )
,
for i ∈ e, l, p, where r > 0 is the constant subjective discount rate. We have
used the shorthand notations: Es· ≡ E·|S0 = s and Ez· ≡ E·|ξ0 = z.By the standard theory of optimal stopping (see e.g. Chapter 1 of Peskir
and Shiryaev [2006] and Chapter 10 of Øksendal [2003]), the optimal stopping
times are of the form
τ ∗i = inf t ≥ 0 : Vi(St, ν) = Ui(νSt) , (4.1.3)
τ ∗i = inf t ≥ 0 : Vi(ξt, ν) = Ui(νξt) . (4.1.4)
In this chapter, we analytically derive the value functions and show they
satisfy their associated variational inequalities. Under the GBM model, for
any fixed ν, the value functions Vi(s) ≡ Vi(s, ν), for i ∈ e, l, p, satisfy the
variational inequalities
max
(LS − r)Vi(s), Ui(νs)− Vi(s)
= 0, ∀s ∈ R+, (4.1.5)
for i ∈ e, l, p, where LS is the infinitesimal generator of S defined by
LS =σ2s2
2
d2
ds2+ µs
d
ds. (4.1.6)
CHAPTER 4. RISK AVERSE TIMING 78
Likewise, under the XOU model the value functions Vi(z) ≡ Vi(z, ν), i ∈e, l, p, solve the variational inequalities
max(LX − r)Vi(ex) , Ui(νex)− Vi(ex) = 0, ∀x ∈ R, (4.1.7)
for i ∈ e, l, p, where
LX =η2
2
d2
dx2+ κ(θ − x)
d
dx, (4.1.8)
is the infinitesimal generator of the OU process X (see (4.1.1)). For optimal
stopping problems driven by an XOU process, we find it more convenient to
work with the log-price X.
To better understand the value of the risky asset under optimal liquidation,
we study the certainty equivalent associated with each utility maximization
problem. The certainty equivalent is defined as the guaranteed cash amount
that generates the same utility as the maximal expected utility from optimally
timing to sell the risky asset. Precisely, we define
(GBM) Ci(s, ν) = U−1i
(Vi(s, ν)
), (4.1.9)
(XOU) Ci(z, ν) = U−1i
(Vi(z, ν)
), (4.1.10)
for i ∈ e, l, p, under the GBM and XOU models respectively. Certainty
equivalent gives us a common (cash) unit to compare the values of timing to
sell under different utilities, dynamics, and quantities.
Moreover, the certainty equivalent can shed light on the investor’s optimal
strategy. Indeed, applying (4.1.9) and (4.1.10) to (4.1.3) and (4.1.4) respec-
tively, we obtain an alternative expression for the optimal stopping time under
each model:
τ ∗i = inf t ≥ 0 : Ci(St, ν) = νSt ,
τ ∗i = inf t ≥ 0 : Ci(ξt, ν) = νξt .
CHAPTER 4. RISK AVERSE TIMING 79
In other words, it is optimal for the investor to sell when the certainty equiva-
lent is equals to the total cash amount of νSt or νξt, under the GBM or XOU
model respectively, received from the sale.
4.2 Optimal Timing Strategies
In this section, we present the analytical results and discuss the value functions
and optimal selling strategies under the GBM and XOU models. The methods
of solution and detailed proofs are presented in Section 4.4.
4.2.1 The GBM Model
To prepare for our results for the GBM model, we first consider an increasing
general solution to the ODE:
LSf(s) = rf(s), s ∈ R+, (4.2.1)
with LS defined in (4.1.6). This solution is f(s) = sα with
α =
(1
2− µ
σ2
)+
√(µ
σ2− 1
2
)2
+2r
σ2. (4.2.2)
By inspection, we see that 0 < α < 1 when r < µ, and α ≥ 1 when r ≥ µ.
Theorem 4.2.1. Consider the optimal asset sale problem (4.1.2) under the
GBM model with exponential utility.
(i) If r ≥ µ, then it is optimal to sell immediately, and the value function is
Ve(s, ν) = 1− e−γνs.
(ii) If r < µ, then the value function is given by
Ve(s, ν) =
(1− e−γνae)(ae)−αsα if s ∈ [0, ae),
1− e−γνs if s ∈ [ae,+∞),
CHAPTER 4. RISK AVERSE TIMING 80
where the optimal threshold ae ∈ (0,+∞) is uniquely determined by the
equation
α(eγνae − 1)− γνae = 0. (4.2.3)
The optimal time to sell is
τ ∗e = inf t ≥ 0 : St ≥ ae .
Under the GBM model with exponential utility, the optimal selling strat-
egy can be either trivial or non-trivial. When the subjective discount rate r
equals or exceeds the drift µ of the GBM process, it is optimal to sell immedi-
ately. This is intuitive as the asset net discounting tends to lose value. On the
other hand, when r < µ, the investor should sell when the unit price exceeds
a finite threshold. At the optimal sale time τ ∗e , the investor receives the cash
amount νae from the sale of ν units of S. In other words, ae is the per-unit
price received upon sale, but according to (4.2.3) it varies depending on the
quantity ν and risk aversion parameter γ.
Theorem 4.2.2. Consider the optimal stopping problem (4.1.2) under the
GBM model with log utility. The value function is given by
Vl(s, ν) =
να
αesα if s ∈ [0, al),
log(νs) if s ∈ [al,+∞),
where al := ν−1 exp(α−1) is the unique optimal threshold. The optimal time to
sell is
τ ∗l = inft ≥ 0 : St ≥ al.
With log utility, the optimal strategy is to sell as soon as the unit price of
the risky asset, S, enters the upper interval [al,+∞). Note that the optimal
CHAPTER 4. RISK AVERSE TIMING 81
threshold al is inversely proportional to quantity, so the total cash amount
received upon sale, νal = exp(α−1), remains the same regardless of quantity. In
other words, the log-utility investor is not financially better off by holding more
units of S. Under exponential utility, the optimal selling price is implicitly
defined by (4.2.3) in Theorem 4.2.1 and must be evaluated numerically. In
contrast, the optimal threshold under log utility is fully explicit.
Turning to the value functions, a natural question is whether they preserve
the concavity of the utilities. Indeed, if the investor sells at a pre-determined
fixed time T , then the expected utility W (s) := Ese−rTU(νST )
is concave in
s for any concave utility function U . From Theorem 4.2.1, we see that Ve(s, ν)
is concave in s for all s ∈ R+. On the other hand, Vl(s, ν) is concave in s when
α < 1, but it is neither convex nor concave in s when α ≥ 1. In other words, the
timing option to sell gives rise to the possibility of non-concave value function.
In Figure 4.1, we plot the value functions associated with the exponential and
log utilities when the investor is holding a single unit of the asset. The value
functions dominate the utility functions, and coincide smoothly at the optimal
selling thresholds. In Figure 4.1(b), the value function Vl(s, 1) under log utility
is shown to have two possible shapes. For µ = 0.01 < 0.02 = r, i.e. α < 1,
the value function Vl(s, 1) is convex when s is lower than al, and concave for
s ≥ al. In the other scenario, µ > r, i.e. α > 1, the value function Vl(s, 1) is
concave.
CHAPTER 4. RISK AVERSE TIMING 82
0 0.5 1 1.5 2 2.5 3 3.5 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
s
Ve(s,1)
Ue(s)
(a)
0 1 2 3 4 5 6 7 8−5
−4
−3
−2
−1
0
1
2
3
s
Vl(s,1) (µ = 0.05)
Vl(s,1) (µ = 0.01)
Ul(s)
(b)
Figure 4.1: Value functions smooth-paste the utility function under the GBM
model. (a) The value function Ve(s, 1) dominates the exponential utility Ue(s) (with
γ = 0.5 and µ = 0.05) and coincides for s ≥ ae = 2.5129. (b) The value functions
Vl(s, 1) (with µ = 0.05) and Vl(s, 1) (with µ = 0.01) dominate the log utility Ul(s)
and coincide for s ≥ al = 7.3891 and s ≥ al = 2.1832 respectively. Common
parameters: σ = 0.2, ν = 1, r = 2%.
1 2 3 4 5 6 7 8 9 100
5
10
15
ν
ae (γ = 0.2)ae (γ = 0.5)ae (γ = 1)al
(a)
Figure 4.2: Optimal selling thresholds, ae (with γ = 0.2, 0.5, 1) and al, under the
GBM model vs quantity ν. Parameters: µ = 0.05, σ = 0.2, r = 2%.
CHAPTER 4. RISK AVERSE TIMING 83
Figure 4.2 illustrates the effect of quantity ν on optimal selling thresholds
ae and al under exponential and log utilities respectively. The optimal strategy
under power utility is trivial and thus omitted from the figure. The optimal
threshold ae is decreasing in ν for each fixed risk aversion γ = 0.2, 0.5 and 1.
Moreover, for any fixed quantity, a higher γ lowers the optimal selling price.
The quantity ν effectively scales up the risk aversion to the value νγ instead
of γ. Increase in either of these parameters results in higher risk aversion,
inducing the investor to sell at a lower price. In comparison, the log-utility
optimal threshold al is explicit and inversely proportional to ν, as seen in the
figure.
We conclude this section with a discussion on the optimal liquidation strat-
egy under power utility. First, observe that for any p ∈ (0, 1], the power process
Spt is also a GBM satisfying
dSpt = µSpt dt+ σSpt dBt,
with new parameters
µ = pµ+1
2p(p− 1)σ2 and σ = pσ.
Then, the process (e−rtSpt )t≥0 is a submartingale (resp. supermartingale) if
µ > r (resp. µ ≤ r). As a result, the optimal timing to sell is trivial, as we
summarize next.
Theorem 4.2.3. Consider the optimal asset sale problem (4.1.2) under the
GBM model with power utility.
(i) If µ ≤ r, then it is optimal to sell immediately, and the value function
Vp(s, ν) = Up(νs).
(ii) If µ > r, then it is optimal to wait indefinitely, and the value function
Vp(s, ν) = +∞.
CHAPTER 4. RISK AVERSE TIMING 84
4.2.2 The XOU Model
In this section, we discuss the optimal asset sale problems under the XOU
model. Recall from Chapter 2 that the classical solutions of the ODE
LXf(x) = rf(x), (4.2.4)
for x ∈ R, are
F (x) ≡ F (x;κ, θ, η, r) :=
∫ ∞0
υrκ−1e
√2κη2
(x−θ)υ−υ2
2 dυ, (4.2.5)
G(x) ≡ G(x;κ, θ, η, r) :=
∫ ∞0
υrκ−1e
√2κη2
(θ−x)υ−υ2
2 dυ. (4.2.6)
Alternatively, the functions F and G can be expressed as
F (x) = eκ
2η2(x−θ)2
D−r/κ
(√2κ
η2(θ − x)
),
and
G(x) = eκ
2η2(x−θ)2
D−r/κ
(√2κ
η2(x− θ)
),
where Dv(·) is the parabolic cylinder function or Weber function (see Erdelyi
et al. [1953]). The strictly positive and convex function F plays a central role
in the solution of the optimal asset sale problems under the XOU model.
Theorem 4.2.4. Under an XOU model with exponential utility, the optimal
asset sale problem admits the solution
Ve(z, ν) =
KF (log(z)) if z ∈ [0, ebe),
1− e−γνz if z ∈ [ebe ,+∞),
(4.2.7)
with the constant
K =1− exp
(−γνebe
)F (be)
> 0.
The critical log-price level be ∈ (−∞,+∞) satisfies(1− exp
(−γνebe
))F ′(be) = γνebe exp
(−γνebe
)F (be). (4.2.8)
CHAPTER 4. RISK AVERSE TIMING 85
The optimal time to sell is
τ ∗e = inft ≥ 0 : ξt ≥ ebe.
According to Theorem 4.2.4, the investor should sell all ν units as soon as
the asset price ξ reaches ebe or above. The optimal price level ebe depends on
both the investor’s risk aversion and quantity, but it stays the same as long as
the product νγ remains unchanged.
Theorem 4.2.5. Under an XOU model with log utility, the optimal asset sale
problem admits the solution
Vl(z, ν) =
DF (log(z)) if z ∈ [0, ebl),
log(νz) if z ∈ [ebl ,+∞),
with the coefficient
D =bl + log(ν)
F (bl)> 0. (4.2.9)
The finite critical log-price level bl is uniquely determined from the equation
F (bl) = (bl + log(ν))F ′(bl). (4.2.10)
The optimal time to sell is
τ ∗l = inft ≥ 0 : ξt ≥ ebl.
In Figure 4.3(d), we see that the optimal unit selling price ebl is decreasing
in ν but when multiplied by the quantity ν, the total cash amount νebl received
from the sale increases.
For the case of power utility, we observe that ξ := ξp is again an XOU
process, satisfying
ξt = eXt , where dXt = κ(θ − Xt) dt+ η dBt, t ≥ 0,
CHAPTER 4. RISK AVERSE TIMING 86
with the new parameters η := pη > 0, and θ := pθ ∈ R. In particular, both
the original long-run mean θ and volatility parameter η have been scaled by a
factor of p, while the speed of mean reversion remains unchanged. Therefore,
the value function admits the separable form:
Vp(z, ν) = supτ∈T
Eze−rτ
νpξpτp
= Up(ν) V (z, 1), (4.2.11)
where
V (z, 1) := supτ∈T
Eze−rτ ξτ
. (4.2.12)
Hence, without loss of generality, the optimal timing to sell can be determined
from the optimal stopping problem in (4.2.12), and the corresponding value
function Vp can be recovered from (4.2.11).
Theorem 4.2.6. Under the XOU model with power utility, the solution to the
optimal asset sale problem is given by
Vp(z, ν) =
MF (p log(z)) if z ∈ [0, ebp),
νp
pzp if z ∈ [ebp ,+∞),
where
M =νpepbp
pF (pbp)> 0.
The critical log-price threshold bp ∈ (−∞,+∞) satisfies the equation
F ′(pbp) = F (pbp), (4.2.13)
where F (x) ≡ F (x;κ, θ, η, r). The optimal asset sale timing is
τ ∗p = inft ≥ 0 : ξt ≥ ebp.
The investor should sell all ν units the first time the asset price reaches
the level ebp . According to (4.2.13), the optimal price level is independent of
quantity ν, as we can see in Figure 4.3(d).
CHAPTER 4. RISK AVERSE TIMING 87
Under the XOU model, the value functions Vi(z, ν), i ∈ e, l, p are not
necessarily concave in z due to the convex nature of F and the timing option
to sell. Let’s inspect the value functions in Figure 4.3. In each of these
three cases, the value function is initially convex before smooth-pasting on the
concave utility.
CHAPTER 4. RISK AVERSE TIMING 88
0.5 1 1.5 2 2.5 3 3.5 40.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
z
Ve(z,1)
Ue(z)
(a)
0.5 1 1.5 2 2.52.6
2.8
3
3.2
3.4
3.6
3.8
4
4.2
4.4
z
Vp(z,1)
Up(z)
(b)
2 2.5 3 3.5 4 4.5 5
0.8
1
1.2
1.4
1.6
1.8
2
z
Vl(z,1)
Ul(z)
(c)
2 4 6 8 10 12 14 16 18 200.5
1
1.5
2
2.5
3
3.5
4
ν
ebe
ebp
ebl
(d)
Figure 4.3: Under the XOU model, (a) the value function Ve(z, 1) dominates the
exponential utility Ue(z) (with γ = 0.5) and coincides for z ≥ ebe = e1.1188 = 3.0612.
(b) The value function Vp(z, 1) dominates the power utility Up(z) (with p = 0.3) and
coincides for z ≥ ebp = e0.3519 = 1.3715. (c) The value function Vl(z, 1) dominates
the log utility Ul(z) and coincides for z ≥ ebl = e1.2227 = 3.3963. (d) Optimal selling
thresholds vs quantity ν. γ = 0.5, p = 0.3. Common Parameters: κ = 0.6, θ = 1, η =
0.2, r = 2%.
CHAPTER 4. RISK AVERSE TIMING 89
In Table 4.1, we summarize the results from Sections 4.2.1 and 4.2.2 and
list the optimal thresholds for all the cases we have discussed. All thresholds,
except al, are implicitly determined by the equations referenced in the table.
The asset price model plays a crucial role in the structure of the optimal strat-
egy. Under the GBM model with exponential utility, immediate liquidation
may be optimal in one scenario regardless of the current asset price. On the
contrary, with the same utility under the XOU model, immediate liquidation is
never optimal and the investor should wait till the asset price rises to level ebe .
With power utility, the GBM model implies a trivial optimal strategy, whereas
the XOU model results in a threshold-type strategy. Lastly, even though both
GBM and XOU price processes lead to non-trivial strategies for log utility, the
optimal threshold al is explicit while ebl must be computed numerically.
Exponential utility Log utility Power utility
GBM 0 / ae in (4.2.3) al := e1/α
ν 0 / +∞
XOU ebe in (4.2.8) ebl in (4.2.10) ebp in (4.2.13)
Table 4.1: Optimal thresholds for asset sale under different models and utili-
ties.
4.3 Certainty Equivalents
Having derived the value functions analytically, we now state as corollaries the
certainty equivalents Ci(s, ν) and Ci(z, ν), i ∈ e, l, p, defined respectively
in (4.1.9) and (4.1.10) under the GBM and XOU models. Furthermore, to
quantify the value gained from waiting to sell the asset compared to immediate
CHAPTER 4. RISK AVERSE TIMING 90
liquidation, we define the optimal liquidation premium under each model:
(GBM) L(s, ν) := Ci(s, ν)− νs,
(XOU) L(z, ν) := Ci(z, ν)− νz,
for i ∈ e, l, p. We will examine the dependence of this premium on the asset
price and quantity.
Corollary 4.3.1. Under the GBM model, the certainty equivalents under dif-
ferent utilities are given as follows:
(1) Exponential utility:
Ce(s, ν) =
−1γ
log(
1− 1−e−γνaeaeα
sα)
if s ∈ [0, ae),
νs if s ∈ [ae,+∞).
(2) Log utility:
Cl(s, ν) =
exp(ναsα
αe
)if s ∈ [0, al),
νs if s ∈ [al,+∞).
(3) Power utility:
Cp(s, ν) =
νs if µ ≤ r,
+∞ if µ > r.
With exponential and log utilities, the certainty equivalents dominate νs –
the value from immediate sale, and they coincide when the asset price exceeds
the corresponding optimal selling thresholds. With power utility, the investor
either sells immediately or waits indefinitely, corresponding to the certainty
equivalents of value νs and +∞, respectively.
The impact of ν on Ce is both direct in its certainty equivalent’s expression,
but also indirect in the derivation of ae. As a result, the relationship between
CHAPTER 4. RISK AVERSE TIMING 91
Ce and ν is rather intricate. In comparison, the explicit formula for the opti-
mal threshold al under log utility facilitates our analysis on the behavior of the
certainty equivalent Cl. Fix any price s, Cl(s, ν) is convex in ν when r ≥ µ.
Consequently, the liquidation premium is maximized at ν = 0. However, when
r < µ, then Cl(s, ν) is concave on the price interval (0, log(
1−αsα
)+ 1) and con-
vex on (log(
1−αsα
)+ 1, exp(α−1)/s). This implies that there exists an optimal
quantity ν∗ ∈ (0, log(
1−αsα
)+1) that maximizes the liquidation premium. This
is useful when the investor can also choose the initial position in S.
Next, we state the certainty equivalents under the XOU model.
Corollary 4.3.2. Under the XOU model, the certainty equivalents under dif-
ferent utilities are given as follows:
(1) Exponential utility:
Ce(z, ν) =
− 1γ
log
[1− 1−exp(−γνebe)
F (be)F (log(z))
]if z ∈ [0, ebe),
νz if z ∈ [ebe ,+∞).
(2) Log utility:
Cl(z, ν) =
exp[bl+log(ν)F (bl)
F (log(z))]
if z ∈ [0, ebl),
νz if z ∈ [ebl ,+∞).
(3) Power utility:
Cp(z, ν) =
[
epbp
F (pbp)F (p log(z))
]1/p
ν if z ∈ [0, ebp),
νz if z ∈ [ebp ,+∞).
For all three utilities, the certainty equivalents are equal to the immediate
sale value, νz, when the asset price z is in the exercise region, where all units
CHAPTER 4. RISK AVERSE TIMING 92
are sold. In addition, we emphasize that both be and bl are dependent on
ν, which can be observed from (4.2.8) and (4.2.10). In contrast, the optimal
log-price threshold under power utility bp is independent of ν. Consequently,
if we consider any fixed z in the continuation region (0, ebp), then the certainty
equivalent Cp(z, ν)−νz is linear and strictly increasing in ν. This is interesting
since under exponential and log utilities, increasing quantity has the effect of
making the investor more risk-averse. In other words, as long as quantity is
large enough, the investor will liquidate everything immediately even if the
current price appears unattractive.
CHAPTER 4. RISK AVERSE TIMING 93
0 1 2 3 4 50
1
2
3
4
5
6
s
γ = 0.2
γ = 0.5
s
(a)
0 2 4 6 8 100
2
4
6
8
10
s
valu
e
Cl(s,1)
s
(b)
0.5 1 1.5 2 2.5 3 3.5 40
0.5
1
1.5
2
2.5
3
3.5
4
z
γ = 0.2
γ = 0.5
γ = 1
z
(c)
0.5 1 1.5 2 2.5 3 3.5 40
0.5
1
1.5
2
2.5
3
3.5
4
z
p = 0.2
p = 0.5
p = 1
z
(d)
Figure 4.4: Certainty equivalent vs price. (a) Ce(s, 1) under the GBM model
(µ = 0.05, σ = 0.2). (b) Cl(s, 1) under the GBM model (µ = 0.05, σ = 0.2). (c)
Ce(z, 1) under the XOU model (κ = 0.6, θ = 1, η = 0.2). (d) Cp(z, 1) under the
XOU model (κ = 0.6, θ = 1, η = 0.2). Note that p := 1 − %, where % is the risk
aversion parameter. Common parameter: r = 2%.
Let us now examine the certainty equivalents’ dependence on the asset
price. Under the GBM model, we plot the certainty equivalents, Ce(s, 1) and
Cl(s, 1), against prices, respectively, in Figures 4.4(a) and 4.4(b), with a single
unit of asset held. The optimal selling strategy for power utility is trivial,
and thus, not presented. From Section 4.2.1, we know that for sufficiently
CHAPTER 4. RISK AVERSE TIMING 94
large s, it is optimal to sell and thus the certainty equivalents will eventually
coincide with s and be linear. Notice that in both Figures 4.4(a) and 4.4(b),
the certainty equivalents are concave for small s and subsequently convex for
large s. In general, the certainty equivalents are neither concave nor convex
functions of asset price, especially since the value functions Vi and Vi, i ∈e, l, p are not necessarily concave.
In Figure 4.4(a), we have also shown Ce(s, 1) for different values of risk
aversion level γ. As the investor becomes more risk-averse, it becomes optimal
to sell the asset earlier. This is reflected by the certainty equivalent’s conver-
gence to the linear price line s at a lower price. Moreover, a less risk-averse
certainty equivalent dominates a more risk-averse one at all prices. Similar
effects of risk aversion is also seen in Figure 4.4(c) for exponential utility and
Figure 4.4(d) for power utility under the XOU model.
510
1520
0.51
1.52
2.5
0
2
4
6
8
zν
(a)
510
1520
0.51
1.52
2.5
0
20
40
60
zν
(b)
Figure 4.5: Liquidation premium L(z, ν) under the XOU model plotted against
quantity ν and price z. (a) Exponential utility with γ = 0.5. (b) Power utility with
p = 0.3. Common parameters: κ = 0.6, θ = 1, η = 0.2, r = 2%.
Figures 4.5(a) and 4.5(b), respectively, illustrate the liquidation premia for
exponential and power utilities under the XOU model. The liquidation pre-
CHAPTER 4. RISK AVERSE TIMING 95
mium for power utility is linear in ν for any fixed value of z, but the exponential
utility liquidation premium is nonlinear. In general, the liquidation premium
vanishes when z is sufficiently high when the asset price is in the exercise re-
gion. As we can see from Figures 4.5(a) and 4.5(b) and Figures 4.4(a)-(d),
the optimal liquidation premium tends to be large and may increase when the
asset price is very low. This suggests that there is a high value of waiting
to sell the asset later if the current price is low. As the asset price rises, the
premium shrinks to zero. The investor finds no value in waiting any longer,
resulting in an immediate sale.
4.4 Methods of Solution and Proofs
In this section, we present the detailed proofs for our analytical results in
Section 4.2, from Theorems 4.2.1 - 4.2.2 for the GBM model to Theorems
4.2.4 - 4.2.6 for the XOU model. Our method of solution is to first construct
candidate solutions using the classical solutions to ODEs (4.2.1) and (4.2.4),
corresponding to the GBM and XOU models respectively, and then verify that
the candidate solutions indeed satisfy the associated variational inequalities
(4.1.5) and (4.1.7).
4.4.1 GBM Model
Proof of Theorem 4.2.1 (Exponential Utility). To prove that the value
functions in Theorem 4.2.1 satisfy the variational inequality in (4.1.5), we
consider the two cases, r ≥ µ and r < µ, separately.
When r ≥ µ, it is optimal to sell immediately. To see this, for any fixed ν
we verify that Ve(s, ν) = 1− e−γνs satisfies (4.1.5). Since Ve(s, ν) = Ue(νs) for
CHAPTER 4. RISK AVERSE TIMING 96
all s ∈ R+, we only need to check that the inequality
(LS − r)(1− e−γνs) = e−γνs(µγνs− σ2γ2ν2s2
2− reγνs + r
)≤ 0, (4.4.1)
holds for all s ∈ R+. First, we observe that the sign of the LHS of (4.4.1)
depends solely on
g(s) := µγνs− σ2γ2ν2s2
2− reγνs + r . (4.4.2)
The first and order second derivatives g are, respectively,
g′ = µγν − σ2γ2ν2s− rγνeγνs, and g′′ = −σ2γ2ν2 − rγ2ν2eγνs,
from which we observe that g is strictly concave on R+. Furthermore, g(0) = 0
and the fact lims→+∞ g(s) = −∞ imply that g has a global maximum. Since
g′+(0) = (µ − r)γν > 0 (resp. < 0) if µ > r (resp. µ < r), the maximum of
g is non-positive if r ≥ µ. As a result, g is non-positive for all s ∈ R+, which
yields inequality (4.4.1), as desired.
For an arbitrary ν > 0, we can view νγ together as the risk aversion
parameter for the exponential utility, and equivalently consider the asset sale
problem with ν = 1 and risk aversion νγ without loss of generality. When
r < µ, we consider a candidate solution Ve of the form Asα, where A > 0 is
a constant to be determined. Recall that α is less than 1 when r < µ, and
hence sα is an increasing concave function. We solve for the optimal threshold
ae and coefficient A from the value-matching and smooth-pasting conditions
Aaαe = Ue(ae) = 1− e−γνae , (4.4.3)
Aαaα−1e = U ′e(ae) = γνe−γνae .
This leads to the following equation satisfied by the optimal threshold ae:
α(eγνae − 1)− γνae = 0. (4.4.4)
CHAPTER 4. RISK AVERSE TIMING 97
We now show that there exists a unique and positive root to (4.4.4). Our
approach involves establishing a relationship between (4.4.4) and (LS− r)(1−e−γνs). To this end, first observe that the exponential utility 1− e−γνs has the
following properties:
lims→0
1− e−γνssβ
= lims→+∞
1− e−γνssα
= 0, (4.4.5)
where
β =
(1
2− µ
σ2
)−√(
µ
σ2− 1
2
)2
+2r
σ2,
and sβ is a decreasing and convex solution to (4.2.1). In addition, we have
Es
∫ ∞0
e−rt∣∣(LS − r)(1− e−γνst)∣∣ dt
= Es∫ ∞
0
e−rt∣∣∣∣e−γνst (µγνst − σ2γ2ν2s2
t
2− reγνst + r
)∣∣∣∣ dt< Es
∫ ∞0
e−rt(
1 +σ2
2+ r
)dt
=
1
r+σ2
2r+ 1 <∞. (4.4.6)
The limits in (4.4.5) and condition (4.4.6) together imply that the function
1− e−γνs admits the following analytic representation:
1− e−γνs =− sβ∫ s
0
ΨS(υ)(LS − r)(1− e−γνυ) dυ (4.4.7)
− sα∫ +∞
s
ΦS(υ)(LS − r)(1− e−γνυ) dυ,
where
ΨS(s) =2sα
σ2s2WS(s), ΦS(s) =
2sβ
σ2s2WS(s),
and
WS(s) =2
√(µ− σ2
2
)2+ 2σ2r
σ2s−
2µ
σ2 > 0, ∀s ∈ R+.
We refer the reader to Section 2 of Zervos et al. [2013] and Chapter 2 of
Borodin and Salminen [2002] for details on the representation.
CHAPTER 4. RISK AVERSE TIMING 98
Next, dividing 1− e−γνs by sα and differentiating in s, we have(1− e−γνs
sα
)′=γνe−γνssα − (1− e−γνs)αsα−1
s2α. (4.4.8)
The crucial step is to recognize that finding the root to the derivative in (4.4.8)
is equivalent to solving (4.4.4). Furthermore, appealing to (4.4.7), the LHS of
(4.4.8) becomes(1− e−γνs
sα
)′= −
(sβ
sα
)′ ∫ s
0
ΨS(υ)(LS − r)(1− e−γνυ
)dυ
− sβ
sαΨS(s)(LS − r)
(1− e−γνs
)+ ΦS(s)(LS − r)(1− e−γνs)
=WS(s)
s2α
∫ s
0
ΨS(υ)(LS − r)(1− e−γνυ
)dυ =
WS(s)
s2αqe(s),
where
qe(s) :=
∫ s
0
ΨS(υ)(LS − r)(1− e−γνυ) dυ.
Since both s2α and WS(s) are strictly positive for s > 0, we conclude that
(4.4.4) is equivalent to the equation qe(ae) = 0. By differentiating, we obtain
q′e(s) = ΨS(s)(LS − r)(1− e−γνs). Since ΨS(s) > 0 for all s ∈ R+, the sign of
q′e(s) depends solely on (LS − r)(1− e−γνs), and thus on g defined in (4.4.2).
The function g is strictly concave on R+. Since g′+(0) = (µ − r)γν > 0, the
maximum of g is strictly positive. This implies that there exists a unique
positive ϕ such that g(ϕ) = 0. Consequently, we have
q′e(s)
> 0 if s < ϕ,
< 0 if s > ϕ.
.
This together with the fact that qe(0) = 0, lead us to conclude that there exists
a unique ae > 0 such that qe(ae) = 0 if and only if lims→+∞ qe(s) < 0. The
latter holds due to the facts:
qe(s) =s2α
WS(s)
(1− e−γνs
sα
)′,
1− e−γνssα
> 0, ∀s ∈ [0,+∞),
lims→+∞
1− e−γνssα
= 0.
CHAPTER 4. RISK AVERSE TIMING 99
Therefore, we conclude that there exists a unique finite root ae to equation
(4.4.4). Furthermore, by the nature of qe we have
ae > ϕ and qe(s) > 0, ∀s < ae.
Finally, from (4.4.3) we deduce that A = (1− e−γνae)a−αe > 0.
Next, we verify the optimality of the candidate solution using the vari-
ational inequality (4.1.5). First, observe that 1 − e−γνs − Ve(s) = 0 on
[ae,+∞), and Ve(s) ≥ 1 − e−γνs for all s ∈ [0, ae). Lastly, the inequality
(LS − r)(1− e−γνs) ≤ 0 follows from
(LS − r)(1− e−γνs) = (LS − r)Asα = 0, for s ∈ [0, ae),
(LS − r)(1− e−γνs) ≤ 0, for s ∈ [ae,+∞).
Hence, Ve(s, ν) given in Theorem 4.2.1 is indeed optimal.
Proof of Theorem 4.2.2 (Log Utility). For any fixed ν, νS follows a
GBM process with the same drift and volatility parameters as S. In other
words, we can reduce the problem to that of selling a single unit of a risky
asset whose price process is S := νS with initial value S0 = s := νs. Therefore,
we construct a candidate solution of the form Vl(s, ν) = Vl(s, 1) = Bsα, where
B is a positive constant. The value-matching and smooth-pasting conditions
are
Baα = Ul(a) = log(a), (4.4.9)
Bαaα−1 = U ′l (a) =1
a.
These equations can be solved explicitly to give a unique solution a = exp(α−1)
and consequently the coefficient B = 1/αe > 0. The optimal aggregate selling
price a translates into the optimal unit selling price al = a/ν = ν−1 exp(α−1).
CHAPTER 4. RISK AVERSE TIMING 100
Next, we show that Vl(s, ν) = Vl(s, 1) ≡ Vl(s) given in Theorem 4.2.2
indeed satisfies the variational inequality
max(LS − r)Vl(s), log(s)− Vl(s) = 0, s ∈ R+,
where LS is the infinitesimal generator of the GBM process S. This is equiv-
alent to showing that Vl(s, ν) in (4.2.2) satisfies the variational inequality
(4.1.5). First, on [0, a) we have (LS − r)Vl(s) = (LS − r)Bsα = 0. Next,
observe that the function
(LS − r) log(s) = −σ2
2+ µ− r log(s),
has a unique root at ϕ = eµ−σ2/2
r such that
(LS − r) log(s)
> 0 if s < ϕ,
< 0 if s > ϕ.
On [a,+∞), we have (LS − r)Vl(s, 1) = (LS − r) log(s). To show that (LS −r)Vl(s, 1) ≤ 0 we need to prove that a > ϕ, or equivalently,
1
α>µ− σ2/2
r.
This follows directly from the definition of α in (4.2.2).
Next, we check that Vl(s, 1) ≥ log(s) for all s ∈ R+. Since Vl(s, 1) = log(s)
on [a,+∞), it remains to show that log(s) ≤ Vl(s, 1) on [0, a). Using (4.4.9),
the desired inequality is equivalent to
log(s)
sα≤ log(a)
aα. (4.4.10)
Differentiating the left-hand side, we get(log(s)
sα
)′=sα−1 − αsα−1 log(s)
s2α=
1− α log(s)
sα+1.
The function log(s)sα
is strictly increasing for s < exp(α−1). Hence, inequality
(4.4.10) follows.
CHAPTER 4. RISK AVERSE TIMING 101
4.4.2 XOU Model
Proof of Theorem 4.2.4 (Exponential Utility). Recall that the func-
tions F and G (see (4.2.5) and (4.2.6)) are respectively increasing and de-
creasing. Since the exponential utility Ue is strictly increasing, we postulate
that the solution to the variational inequality (4.1.7) is of the form KF (x),
where K is a positive coefficient to be determined. By grouping ν with γ, the
problem of selling ν units of the risky asset can be reduced to that of selling
a single unit. The value-matching and smooth-pasting conditions are
KF (be) = 1− exp(−γνebe
), (4.4.11)
KF ′(be) = γνebe exp(−γνebe
). (4.4.12)
Using (4.4.11), we have K = (1− exp(−γνebe
))F (be)
−1 > 0. Combining
(4.4.11) and (4.4.12), we obtain (4.2.8) for be.
Next, we want to establish that
Ex∫ ∞
0
e−rt∣∣(LX − r) (1− exp
(−γνeXt
))∣∣ dt <∞. (4.4.13)
First, using (4.1.8) we compute
(LX − r) (1− exp (−γνex)) =η2
2γνex exp (−γνex) (1− γνex)
+ κ(θ − x)γνex exp (−γνex)
− r (1− exp (−γνex)) .
Then, for any T > 0,
Ex∫ T
0
e−rt∣∣(LX − r) (1− exp
(−γνeXt
))∣∣ dt< Ex
∫ T
0
e−rt(η2
2
(1 + γνeXt
)+ κ|θ|+ κ|Xt|+ r
)dt
=
∫ T
0
e−rt(η2
2+η2γν
2ExeXt
+ κ|θ|+ κEx |Xt|+ r
)dt
<η2
2r+η2γν
2re|x|+|θ|+
η2
4κ +κ|θ|r
+κ
r
(√η2
πκ+ |x|+ |θ|
)+ 1,
CHAPTER 4. RISK AVERSE TIMING 102
where we have used the fact that |Xt| conditioned on X0 = x has a folded
normal distrbution. Furthermore, since this bound is time-independent and
finite, we deduce that (4.4.13) is indeed true. We follow the arguments in
Section 2 of Zervos et al. [2013] to obtain the representation
1− exp (−γνex) =−G(x)
∫ x
−∞ΨX(υ)(LX − r)(1− exp (−γνeυ)) dυ
− F (x)
∫ +∞
x
ΦX(υ)(LX − r)(1− exp (−γνeυ)) dυ,
(4.4.14)
where
ΨX(x) :=2F (x)
η2WX(x), ΦX(x) :=
2G(x)
η2WX(x), (4.4.15)
and
WX(x) = F ′(x)G(x)− F (x)G′(x) > 0, ∀x ∈ R.
To see the connection between (4.4.14) and (4.2.8), we divide both sides of
(4.4.14) by F (x) and differentiate with respect to x, and get the derivative(1− exp (−γνex)
F (x)
)′=γνex exp (−γνex)F (x)− (1− exp (−γνex))F ′(x)
F 2(x)(4.4.16)
= −(G(x)
F (x)
)′ ∫ x
−∞ΨX(υ)(LX − r)
(1− e−γνeυ
)dυ
− G(x)
F (x)ΨX(x)(LX − r)
(1− e−γνex
)+ ΦX(x)(LX − r)
(1− e−γνex
)=WX(x)
F 2(x)
∫ x
−∞ΨX(υ)(LX − r)
(1− e−γνeυ
)dυ
=WX(x)
F 2(x)qe(x), (4.4.17)
where
qe(x) :=
∫ x
−∞ΨX(υ)(LX − r) (1− exp (−γνeυ)) dυ. (4.4.18)
CHAPTER 4. RISK AVERSE TIMING 103
By comparing (4.2.8) to the numerator on the RHS of (4.4.16), and given that
WX(x)F 2(x)
in (4.4.17) is strictly positive, we see that the equation satisfied by be
in (4.2.8) is equivalent to qe(be) = 0. Therefore, our goal is to show that qe(x)
has a unique and finite root.
Differentiating (4.4.18) with respect to x yields
q′e(x) = ΨX(x)(LX − r) (1− exp (−γνex)) .
This implies that the sign of q′e depends solely on (LX − r) (1− exp (−γνex)) .We proceed to show that q′e(x) has a unique root. To facilitate computation,
we define a new function
h(x) :=q′e(x)
ΨX(x)× exp (γνex)
γνex= (LX − r) (1− exp (−γνex)) exp (γνex)
γνex
=η2
2(1− γνex) + κ(θ − x)
− r(
exp (γνex)
γνex− 1
γνex
).
Since h is obtained through dividing and multiplying q′e by strictly positive
terms, any root of q′e must also be a root of h and vice-versa.
To find the root of h, we solve
−η2γν
2ex − r
γνe−x (exp (γνex)− 1) = κx− κθ − η2γν
2. (4.4.19)
The RHS of (4.4.19) is a strictly increasing linear function. As for the LHS,
we observe that
limx→+∞
−η2γν
2ex − r
γνe−x (exp (γνex)− 1) = −∞,
limx→−∞
−η2γν
2ex − r
γνe−x (exp (γνex)− 1) = r.
Hence, in order for h to have a unique root, it suffices to show that the LHS
of (4.4.19) is strictly decreasing. Given that r, γ, ν > 0 and ex is strictly
CHAPTER 4. RISK AVERSE TIMING 104
increasing, it remains to show that the function e−x (exp (γνex)− 1) is strictly
increasing for all x ∈ R. The quotient rule gives(exp (γνex)− 1
ex
)′=
exp (γνex) (γνex − 1) + 1
ex.
The numerator exp (γνex) (γνex − 1)+1 goes to +∞ as x goes to +∞ and goes
to 0 as x goes to −∞. Moreover, the derivative of exp (γνex) (γνex − 1) + 1
is γ2ν2e2x exp (γνex) which is strictly positive. This proves that the function
e−x (exp (γνex)− 1) is indeed strictly increasing and as a result, h has a unique
root, denoted by ζ. Finally, observe that
q′e(x) =
> 0 if x < ζ,
< 0 if x > ζ.
(4.4.20)
Combining (4.4.20) with limx→−∞
qe(x) = 0, we now see that a unique root, be,
such that qe(be) = 0, exists if and only if limx→+∞
qe(x) < 0. To examine this
limit, we apply the definition of F to get
qe(x) =F 2(x)
WX(x)
(1− exp (−γνex)
F (x)
)′, (4.4.21)
1− exp (−γνex)F (x)
> 0, ∀x ∈ R, limx→+∞
1− exp (−γνex)F (x)
=1
+∞ = 0.
(4.4.22)
Since qe is strictly decreasing in (ζ,+∞), (4.4.21) and (4.4.22) hold if and only
if limx→+∞
qe(x) < 0. This shows, as desired, that there exists a unique and finite
be such that
γνebe exp(−γνebe
)F (be) =
(1− exp
(−γνebe
))F ′(be).
Moreover, by (4.4.20), we see that
be > ζ and qe(x) > 0, ∀x < be. (4.4.23)
CHAPTER 4. RISK AVERSE TIMING 105
Now, in order to ascertain the optimality of Ve presented in (4.2.7), we
re-express Ve in terms of the variable x, and show that it satisfies the following
variational inequality:
max(LX − r)Ve(ex), (1− exp (−γνex))− Ve(ex) = 0, ∀x ∈ R.
Indeed, this follows from direct substitution. First, on [be,+∞), we have
(1− exp (−γνex))− Ve(ex) = (1− exp (−γνex))− (1− exp (−γνex)) = 0.
On (−∞, be), we apply (4.4.21) and (4.4.23) to conclude that
(1− exp (−γνex))− Ve(ex) = (1− exp (−γνex))− 1− exp(−γνebe
)F (be)
F (x) ≤ 0.
Next, we verify that (LX − r)Ve(ex) ≤ 0, ∀x ∈ R. To this end, we have
(LX − r)Ve(ex) = (LX − r)KF (x) = 0, on (−∞, be),
(LX − r)Ve(ex) = (LX − r) (1− exp (−γνex)) ≤ 0, on [be,+∞),
as a consequence of (4.2.4) and (4.4.20). Hence, we conclude the optimality of
Ve in (4.2.7).
Proof of Theorem 4.2.5 (Log Utility). Since log(νξ) = X+log(ν) where
X is an OU process, the optimal asset sale problem can be viewed as that under
an OU process with a linear utility and a transaction cost (resp. reward) of
value log(ν) if ν < 1 (resp. ν > 1) (see Leung and Li [2015]).
The functions F and G given in (4.2.5) and (4.2.6) are respectively strictly
increasing and decreasing functions. For any given ν, X + log(ν) is also a
strictly increasing function. This prompts us to postulate a solution to the
variational inequality (4.1.7) of the form DF (x) where D > 0 is a constant to
be determined. Consequently, the optimal log-price thershold bl is determined
from the following value-matching and smooth-pasting conditions:
DF (bl) = bl + log(ν), and DF ′(bl) = 1.
CHAPTER 4. RISK AVERSE TIMING 106
Combining these equations leads to (4.2.9) and (4.2.10). Straightforward com-
putation yields
(LX − r)(x+ log(ν)) = −(κ+ r)x+ κθ − r log(ν),
which is a strictly decreasing linear function with a unique root `. For any
T > 0,
Ex∫ T
0
e−rt∣∣(LX − r)(Xt + log(ν))
∣∣ dt< Ex
∫ T
0
e−rt ((κ+ r)|Xt|+ κ|θ|+ r| log(ν)|) dt
=
∫ T
0
e−rt ((κ+ r)Ex |Xt|+ κ|θ|+ r| log(ν)|) dt
<κ+ r
r
(√η2
πκ+ |x|+ |θ|
)+κ|θ|r
+ | log(ν)|,
which implies that
Ex∫ ∞
0
e−rt∣∣(LX − r)(Xt + log(ν))
∣∣ dt <∞.
With this, we follow the arguments in Section 2 of Zervos et al. [2013] to obtain
the representation
x+ log(ν) =−G(x)
∫ x
−∞ΨX(υ)(LX − r)(υ + log(ν)) dυ (4.4.24)
− F (x)
∫ +∞
x
ΦX(υ)(LX − r)(υ + log(ν)) dυ,
where ΨX and ΦX are as defined in (4.4.15). We relate (4.4.24) to (4.2.10) by
first dividing both sides of (4.4.24) by F (x) and differentiating with respect to
CHAPTER 4. RISK AVERSE TIMING 107
x. This yields(x+ log(ν)
F (x)
)′=F (x)− F ′(x)(x+ log(ν))
F 2(x)(4.4.25)
= −(G(x)
F (x)
)′ ∫ x
−∞ΨX(υ)(LX − r)(υ + log(ν)) dυ
− G(x)
F (x)ΨX(x)(LX − r)(x+ log(ν)) + ΦX(x)(LX − r)(x+ log(ν))
=WX(x)
F 2(x)
∫ x
−∞ΨX(υ)(LX − r)(υ + log(ν)) dυ =
WX(x)
F 2(x)ql(x),
where
ql(x) :=
∫ x
−∞ΨX(υ)(LX − r)(υ + log(ν)) dυ.
Comparing (4.2.10) and the RHS of (4.4.25), along with the facts that F > 0
and WX > 0, we see that solving equation (4.2.10) for the log-price threshold
is equivalent to solving
ql(x) = 0.
Direct differentiation yields that
q′l(x) = ΨX(x)(LX − r)(x+ log(ν))
> 0 if x < `,
< 0 if x > `.
(4.4.26)
The fact that limx→−∞ ql(x) = 0 implies that there exists a unique bl such
that ql(bl) = 0 if and only if limx→+∞ ql(x) < 0. By the definition of F , we
have
ql(x) =F 2(x)
WX(x)
(x+ log(ν)
F (x)
)′,
x+ log(ν)
F (x)> 0, ∀x > − log(ν),(4.4.27)
limx→+∞
x+ log(ν)
F (x)= 0.
Given that ql is strictly decreasing in (`,+∞), we conclude that in order for
(4.4.27) to hold, we must have limx→+∞ ql(x) < 0. This means that there
CHAPTER 4. RISK AVERSE TIMING 108
exists a unique and finite bl such that (bl + log(ν))F ′(bl) = F (bl). Moreover,
given (4.4.26), we have
bl > ` and ql(x) > 0, ∀x < bl. (4.4.28)
Furthermore, since both F and F ′ are strictly positive, bl + log(ν) must also
be positive.
Lastly, we need to check that the following variational inequality holds for
any fixed ν:
max(LX − r)Vl(ex), (x+ log(ν))− Vl(ex) = 0, ∀x ∈ R.
To begin, on the interval [bl,+∞), we have (x+log(ν))−Vl(ex) = (x+log(ν))−(x+ log(ν)) = 0. Next, on the region (−∞, bl), we have
(x+ log(ν))− Vl(ex) = (x+ log(ν))− bl + log(ν)
F (bl)F (x) ≤ 0
since the function x+log(ν)F (x)
is increasing on (−∞, bl) due to (4.4.27) and (4.4.28).
Also, we note that
(LX − r)Vl(ex) = (LX − r)DF (x) = 0, on (−∞, bl),
(LX − r)Vl(ex) = (LX − r)(x+ log(ν)) ≤ 0, on [bl,+∞).
The latter inequality is true due to (4.4.26) and (4.4.28). Vl defined in Theorem
4.2.5 is therefore the optimal solution to the optimal asset sale problem under
log utility.
Proof of Theorem 4.2.6 (Power Utility). Since the powered XOU pro-
cess, ξp, is still an XOU process, the asset sale problem is an optimal stopping
problem driven by an XOU process, which has been solved by the author’s
prior work; see Theorem 3.1.1 of Leung et al. [2015]. The theorem is also
presented in this dissertation as Theorem 2.2.2 of Chapter 2. Therefore, we
omit the proof.
BIBLIOGRAPHY 109
Bibliography
Abramowitz, M. and Stegun, I. (1965). Handbook of Mathematical Functions:
with Formulas, Graphs, and Mathematical Tables, volume 55. Dover Publi-
cations.
Acworth, W. (2016). 2015 annual survey: Global derivatives volume. [Online;
posted 15-March-2016].
Alili, L., Patie, P., and Pedersen, J. (2005). Representations of the first hit-
ting time density of an Ornstein-Uhlenbeck process. Stochastic Models,
21(4):967–980.
Anthony, M. and MacDonald, R. (1998). On the mean-reverting properties
of target zone exchange rates: Some evidence from the ERM. European
Economic Review, 42(8):1493–1523.
Bali, T. G. and Demirtas, K. O. (2008). Testing mean reversion in financial
market volatility: Evidence from S&P 500 index futures. Journal of Futures
Markets, 28(1):1–33.
Balvers, R., Wu, Y., and Gilliland, E. (2000). Mean reversion across national
stock markets and parametric contrarian investment strategies. The Journal
of Finance, 55(2):745–772.
Benth, F. E. and Karlsen, K. H. (2005). A note on Merton’s portfolio selection
BIBLIOGRAPHY 110
problem for the Schwartz mean-reversion model. Stochastic Analysis and
Applications, 23(4):687–704.
Bessembinder, H., Coughenour, J. F., Seguin, P. J., and Smoller, M. M. (1995).
Mean reversion in equilibrium asset prices: Evidence from the futures term
structure. The Journal of Finance, 50(1):361–375.
Borodin, A. and Salminen, P. (2002). Handbook of Brownian Motion: Facts
and Formulae. Birkhauser, 2nd edition.
Brennan, M. J. and Schwartz, E. S. (1990). Arbitrage in stock index futures.
Journal of Business, 63(1):S7–S31.
Carmona, J. and Leon, A. (2007). Investment option under CIR interest rates.
Finance Research Letters, 4(4):242–253.
Cartea, A., Jaimungal, S., and Penalva, J. (2015). Algorithmic and High-
Frequency Trading. Cambridge University Press, Cambridge, England.
Casassus, J. and Collin-Dufresne, P. (2005). Stochastic convenience yield im-
plied from commodity futures and interest rates. The Journal of Finance,
60(5):2283–2331.
Chiu, M. C. and Wong, H. Y. (2012). Dynamic cointegrated pairs trading:
Time-consistent mean-variance strategies. Technical report, working paper.
Cox, J. C., Ingersoll, J., and Ross, S. A. (1981). The relation between forward
prices and futures prices. Journal of Financial Economics, 9(4):321–346.
Cox, J. C., Ingersoll, J. E., and Ross, S. A. (1985). A theory of the term
structure of interest rates. Econometrica, 53(2):385–408.
Dai, M., Zhong, Y., and Kwok, Y. K. (2011). Optimal arbitrage strategies
on stock index futures under position limits. Journal of Futures Markets,
31(4):394–406.
BIBLIOGRAPHY 111
Detemple, J. and Osakwe, C. (2000). The valuation of volatility options.
European Finance Review, 4(1):21–50.
Ekstrom, E., Lindberg, C., and Tysk, J. (2011). Optimal liquidation of a pairs
trade. In Nunno, G. D. and Øksendal, B., editors, Advanced Mathematical
Methods for Finance, chapter 9, pages 247–255. Springer-Verlag.
Ekstrom, E. and Vaicenavicius, J. (2016). Optimal liquidation of an asset
under drift uncertainty. Working Paper.
Elton, E. J., Gruber, M. J., Brown, S. J., and Goetzmann, W. N. (2009).
Modern Portfolio Theory and Investment Analysis. Wiley, 8th edition.
Engel, C. and Hamilton, J. D. (1989). Long swings in the exchange rate: Are
they in the data and do markets know it? Technical report, National Bureau
of Economic Research.
Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G. (1953). Higher
Transcendental Functions, volume 2. McGraw-Hill.
Evans, J., Henderson, V., and Hobson, D. (2008). Optimal timing for an
indivisible asset sale. Mathematical Finance, 18(4):545–567.
Ewald, C.-O. and Wang, W.-K. (2010). Irreversible investment with
Cox-Ingersoll-Ross type mean reversion. Mathematical Social Sciences,
59(3):314–318.
Feller, W. (1951). Two singular diffusion problems. The Annals of Mathemat-
ics, 54(1):173–182.
Geman, H. (2007). Mean reversion versus random walk in oil and natural gas
prices. In Fu, M. C., Jarrow, R. A., Yen, J.-Y. J., and Elliot, R. J., edi-
tors, Advances in Mathematical Finance, Applied and Numerical Harmonic
Analysis, pages 219–228. Birkhuser Boston.
BIBLIOGRAPHY 112
Going-Jaeschke, A. and Yor, M. (2003). A survey and some generalizations of
Bessel processes. Bernoulli, 9(2):313–349.
Gorton, G. B., Hayashi, F., and Rouwenhorst, K. G. (2013). The fundamentals
of commodity futures returns. Review of Finance, 17(1):35–105.
Gropp, J. (2004). Mean reversion of industry stock returns in the US, 1926–
1998. Journal of Empirical Finance, 11(4):537–551.
Grubichler, A. and Longstaff, F. (1996). Valuing futures and options on volatil-
ity. Journal of Banking and Finance, 20(6):985–1001.
Guo, X. and Zervos, M. (2010). π options. Stochastic processes and their
applications, 120(7):1033–1059.
Henderson, V. (2007). Valuing the option to invest in an incomplete market.
Mathematics and Financial Economics, 1(2):103–128.
Henderson, V. (2012). Prospect theory, liquidation, and the disposition effect.
Management Science, 58(2):445–460.
Henderson, V. and Hobson, D. (2011). Optimal liquidation of derivative port-
folios. Mathematical Finance, 21(3):365–382.
Heston, S. L. (1993). A closed form solution for options with stochastic volatil-
ity with applications to bond and currency options. Review of Financial
Studies, 6:327–343.
Irwin, S. H., Zulauf, C. R., and Jackson, T. E. (1996). Monte Carlo anal-
ysis of mean reversion in commodity futures prices. American Journal of
Agricultural Economics, 78(2):387–399.
Ito, K. and McKean, H. (1965). Diffusion processes and their sample paths.
Springer Verlag.
BIBLIOGRAPHY 113
Jurek, J. W. and Yang, H. (2007). Dynamic portfolio selection in arbitrage.
Working Paper, Princeton University.
Karlin, S. and Taylor, H. M. (1981). A Second Course in Stochastic Processes,
volume 2. Academic Press.
Kong, H. T. and Zhang, Q. (2010). An optimal trading rule of a mean-reverting
asset. Discrete and Continuous Dynamical Systems, 14(4):1403 – 1417.
Larsen, K. S. and Sørensen, M. (2007). Diffusion models for exchange rates in
a target zone. Mathematical Finance, 17(2):285–306.
Lebedev, N. (1972). Special Functions & Their Applications. Dover Publica-
tions.
Leung, T., Li, J., Li, X., and Wang, Z. (2016). Speculative futures trading
under mean reversion. Asia-Pacific Financial Markets, pages 1–24.
Leung, T. and Li, X. (2015). Optimal mean reversion trading with transaction
costs and stop-loss exit. International Journal of Theoretical & Applied
Finance, 18(3):15500.
Leung, T., Li, X., and Wang, Z. (2014). Optimal starting–stopping and switch-
ing of a CIR process with fixed costs. Risk and Decision Analysis, 5(2):149–
161.
Leung, T., Li, X., and Wang, Z. (2015). Optimal multiple trading times
under the exponential OU model with transaction costs. Stochastic Models,
31(4):554–587.
Leung, T. and Liu, P. (2012). Risk premia and optimal liquidation of
credit derivatives. International Journal of Theoretical & Applied Finance,
15(8):1250059.
BIBLIOGRAPHY 114
Leung, T. and Ludkovski, M. (2012). Accounting for risk aversion in derivatives
purchase timing. Mathematics & Financial Economics, 6(4):363–386.
Leung, T. and Shirai, Y. (2015). Optimal derivative liquidation timing
under path-dependent risk penalties. Journal of Financial Engineering,
2(1):1550004.
Leung, T. and Wang, Z. (2016). Optimal risk-averse timing of an asset sale:
Trending vs mean-reverting price dynamics. Working Paper.
Lu, Z. and Zhu, Y. (2009). Volatility components: The term structure dynam-
ics of VIX futures. Journal of Futures Markets, 30(3):230–256.
Malliaropulos, D. and Priestley, R. (1999). Mean reversion in Southeast Asian
stock markets. Journal of Empirical Finance, 6(4):355–384.
Mencıa, J. and Sentana, E. (2013). Valuation of VIX derivatives. Journal of
Financial Economics, 108(2):367–391.
Metcalf, G. E. and Hassett, K. A. (1995). Investment under alternative re-
turn assumptions comparing random walks and mean reversion. Journal of
Economic Dynamics and Control, 19(8):1471–1488.
Monoyios, M. and Sarno, L. (2002). Mean reversion in stock index futures
markets: a nonlinear analysis. The Journal of Futures Markets, 22(4):285–
314.
Moskowitz, T. J., Ooi, Y. H., and Pedersen, L. H. (2012). Time series momen-
tum. Journal of Financial Economics, 104(2):228–250.
Øksendal, B. (2003). Stochastic Differential Equations: an Introduction with
Applications. Springer.
BIBLIOGRAPHY 115
Pedersen, J. L. and Peskir, G. (2016). Optimal mean-variance selling strate-
gies. Mathematics and Financial Economics, 10(2):203–220.
Peskir, G. and Shiryaev, A. N. (2006). Optimal Stopping and Free-boundary
problems. Birkhauser-Verlag, Lectures in Mathematics, ETH Zurich.
Poterba, J. M. and Summers, L. H. (1988). Mean reversion in stock prices:
Evidence and implications. Journal of Financial Economics, 22(1):27–59.
Ribeiro, D. R. and Hodges, S. D. (2004). A two-factor model for commodity
prices and futures valuation. EFMA 2004 Basel Meetings Paper.
Rogers, L. and Williams, D. (2000). Diffusions, Markov Processes and Mar-
tingales, volume 2. Cambridge University Press, UK, 2nd edition.
Schwartz, E. (1997). The stochastic behavior of commodity prices: Implica-
tions for valuation and hedging. The Journal of Finance, 52(3):923–973.
Shiryaev, A., Xu, Z., and Zhou, X. (2008). Thou shalt buy and hold. Quanti-
tative Finance, 8(8):765–776.
Song, Q., Yin, G., and Zhang, Q. (2009). Stochastic optimization methods
for buying-low-and-selling-high strategies. Stochastic Analysis and Applica-
tions, 27(3):523–542.
Song, Q. and Zhang, Q. (2013). An optimal pairs-trading rule. Automatica,
49(10):3007–3014.
Tourin, A. and Yan, R. (2013). Dynamic pairs trading using the stochastic
control approach. Journal of Economic Dynamics and Control, 37(10):1972–
1981.
Wang, Z. and Daigler, R. T. (2011). The performance of VIX option pricing
models: Empirical evidence beyond simulation. Journal of Futures Markets,
31(3):251–281.
BIBLIOGRAPHY 116
Wilmott, P., Howison, S., and Dewynne, J. (1995). The Mathematics of Fi-
nancial Derivatives: A Student Introduction. Cambridge University Press,
1st edition.
Zervos, M., Johnson, T., and Alazemi, F. (2013). Buy-low and sell-high in-
vestment strategies. Mathematical Finance, 23(3):560–578.
Zhang, H. and Zhang, Q. (2008). Trading a mean-reverting asset: Buy low
and sell high. Automatica, 44(6):1511–1518.
Zhang, J. E. and Zhu, Y. (2006). VIX futures. Journal of Futures Markets,
26(6):521–531.
Zhu, S.-P. and Lian, G.-H. (2012). An analytical formula for VIX futures and
its applications. Journal of Futures Markets, 32(2):166–190.
APPENDIX A. APPENDIX FOR GBM EXAMPLE 117
Appendix A
Appendix for GBM Example
Let S be a geometric Brownian motion with drift and volatility parameters
(µ, σ). In this case, the optimal exit problem is trivial. Indeed, if µ > r, then
V(s) := supτ∈T
Ese−rτ (Sτ − cs)
≥ supt≥0
(Ese−rtSt − e−rtcs
)≥ sup
t≥0se(µ−r)t − cs = +∞.
Therefore, it is optimal to take τ = +∞ and the value function is infinite.
If µ = r, then the value function is given by
V(s) = supt≥0
supτ∈T
Ese−r(τ∧t)(Sτ∧t − cs)
= s− cs inft≥0
infτ∈T
Ese−r(τ∧t) = s, (A.0.1)
where the second equality follows from the optional sampling theorem and that
(e−rtSt)t≥0 is a martingale. Again, the optimal value is achieved by choosing
τ = +∞, but V(s) is finite in (A.0.1).
On the other hand, if µ < r, then we have a non-trivial solution and exit
timing:
V(s) =
(
csη−1
)1−η (sη
)ηif x < s∗,
s− cs if x ≥ s∗,
APPENDIX A. APPENDIX FOR GBM EXAMPLE 118
where
η =
√2rσ2 + (µ− 1
2σ2)2 − (µ− 1
2σ2)
σ2and s∗ =
csη
η − 1> cs.
Therefore, it is optimal to liquidate as soon as S reaches level s∗. However, it
is optimal not to enter because sups∈R+(V(s)− s− cb) ≤ 0, giving a zero value
for the entry timing problem. Guo and Zervos [2010] provide a detailed study
on this problem and its variation in the context of π options.
APPENDIX B. APPENDIX FOR CHAPTER 2 119
Appendix B
Appendix for Chapter 2
B.1 Proof of Lemma 2.2.5 (Bounds of J ξ and
V ξ)
By definition, both Jξ(x) and V ξ(x) are nonnegative. Using Dynkin’s for-
mula,we have
Exe−rτneXτn − Exe−rνneXνn = Ex∫ τn
νn
e−rt(L − r)eXtdt
= Ex∫ τn
νn
e−rteXt(σ2
2+ µθ − r − µXt
)dt
.
The function ex(σ2
2+ µθ − r − µx
)is bounded above on R. Let M be an
upper bound, it follows that
Exe−rτneXτn − Exe−rνneXνn ≤MEx∫ τn
νn
e−rtdt
.
APPENDIX B. APPENDIX FOR CHAPTER 2 120
Since ex − cs ≤ ex and ex + cb ≥ ex, we have
Ex
∞∑n=1
[e−rτnhξs(Xτn)− e−rνnhξb(Xνn)]
≤∞∑n=1
(Ee−rτneXτn − Exe−rνneXνn
)≤
∞∑n=1
MEx∫ τn
νn
e−rtdt
≤M
∫ ∞0
e−rtdt =M
r:= C1,
which implies that 0 ≤ Jξ(x) ≤ C1. Similarly,
Ex
e−rτ1hξs(Xτ1) +
∞∑n=2
[e−rτnhξs(Xτn)− e−rτnhξb(Xτn)]
≤ C1 + Exe−rτ1hξb(Xτ1).
Letting ν1 = 0 and using Dynkin’s formula again, we have
Exe−rτ1eXτ1 − ex ≤M
r.
This implies that
V ξ(x) ≤ C1 + ex +M
r:= ex + C2.
B.2 Proof of Lemma 2.2.12 (Bounds of Jχ and
V χ)
By definition, both Jχ(y) and V χ(y) are nonnegative. Using Dynkin’s for-
mula,we have
Eye−rτnYτn − Eye−rνnYνn = Ey∫ τn
νn
e−rt(Lχ − r)Ytdt
= Ey∫ τn
νn
e−rt (µθ − (r + µ)Yt) dt
.
For y ≥ 0, the function µθ − (r + µ)y is bounded by µθ. It follows that
Eye−rτnYτn − Eye−rνnYνn ≤ µθEy∫ τn
νn
e−rtdt
.
APPENDIX B. APPENDIX FOR CHAPTER 2 121
Since y − cs ≤ y and y + cb ≥ y, we have
Ey
∞∑n=1
[e−rτnhχs (Yτn)− e−rνnhχb (Yνn)]
≤∞∑n=1
(Ee−rτnYτn − Eye−rνnYνn
)≤
∞∑n=1
µθEy∫ τn
νn
e−rtdt
≤ µθ
∫ ∞0
e−rtdt =µθ
r.
This implies that 0 ≤ Jχ(y) ≤ µθr
. Similarly,
Ey
e−rτ1hχs (Yτ1) +
∞∑n=2
[e−rτnhχs (Yτn)− e−rτnhχb (Yτn)]
≤ µθ
r+ Ey
e−rτ1hχb (Yτ1)
.
Letting ν1 = 0 and using Dynkin’s formula again, we have
Eye−rτ1Yτ1 − y ≤µθ
r.
This implies that
V χ(y) ≤ µθ
r+ y +
µθ
r:= y +
2µθ
r.
APPENDIX C. APPENDIX FOR CHAPTER 3 122
Appendix C
Appendix for Chapter 3
C.1 Numerical Implementation
We apply a finite difference method to compute the optimal boundaries in
Figures 3.2, 3.3 and 3.4. The operators L(i), i ∈ 1, 2, 3, defined in (3.3.2)-
(3.3.3) correspond to the OU, CIR, and XOU models, respectively. To capture
these models, we define the generic differential operator
L· := −r ·+∂·∂t
+ ϕ(s)∂·∂s
+σ2(s)
2
∂2·∂s2
,
then the variational inequalities (3.3.4), (3.3.5), (3.3.6), (3.3.7) and (3.3.8)
admit the same form as the following variational inequality problem:
Lg(t, s) ≤ 0, g(t, s) ≥ ξ(t, s), (t, s) ∈ [0, T )× R+,
(Lg(t, s))(ξ(t, s)− g(t, s)) = 0, (t, s) ∈ [0, T )× R+,
g(T , s) = ξ(T , s), s ∈ R+.
Here, g(t, s) represents the value functions V(t, s), J (t, s), −U(t, s), K(t, s), or
P(t, s). The function ξ(t, s) represents f(t, s;T )−c, (V(t, s)−(f(t, s;T )+ c))+,
−(f(t, s;T )+c), (f(t, s;T )−c)−U(t, s))+, or maxA(t, s),B(t, s). The futures
APPENDIX C. APPENDIX FOR CHAPTER 3 123
price f(t, s;T ), with T ≤ T , is given by (3.1.1), (3.1.4), and (3.1.10) under the
OU, CIR, and XOU models, respectively.
We now consider the discretization of the partial differential equation Lg(t, s) =
0, over an uniform grid with discretizations in time (δt = TN
), and space
(δs = SmaxM
). We apply the Crank-Nicolson method, which involves the finite
difference equation:
−αigi−1,j−1 + (1− βi)gi,j−1 − γigi+1,j−1 = αigi−1,j + (1 + βi)gi,j + γigi+1,j,
where
gi,j = g(jδt, iδs), ξi,j = ξ(jδt, iδs), ϕi = ϕ(iδs), σi = σ(iδs).
αi =δt
4δs
(σ2i
δs− ϕi
), βi = −δt
2
(r +
σ2i
(δs)2
), γi =
δt
4δs
(σ2i
δs+ ϕi
),
for i = 1, 2, ...,M−1 and j = 1, 2, ..., N−1. The system to be solved backward
in time is
M1gj−1 = rj,
where the right-hand side is
rj = M2gj + α1
g0,j−1 + g0,j
0
...
0
+ γM−1
0
...
0
gM,j−1 + gM,j,
,
APPENDIX C. APPENDIX FOR CHAPTER 3 124
and
M1 =
1− β1 −γ1
−α2 1− β2 −γ2
−α3 1− β3 −γ3
. . . . . . . . .
−αM−2 1− βM−2 −γM−2
−αM−1 1− βM−1
,
M2 =
1 + β1 γ1
α2 1 + β2 γ2
α3 1 + β3 γ3
. . . . . . . . .
αM−2 1 + βM−2 γM−2
αM−1 1 + βM−1
,
gj =
g1,j, g2,j, . . . , gM−1,j
T .This leads to a sequence of stationary complementarity problems. Hence, at
APPENDIX C. APPENDIX FOR CHAPTER 3 125
each time step j ∈ 1, 2, . . . , N − 1, we need to solve
M1gj−1 ≥ rj,
gj−1 ≥ ξj−1,
(M1gj−1 − rj)T (ξj−1 − gj−1) = 0.
To solve the optimal problem, our algorithm enforces the constraint explicitly
as follows
gnewi,j−1 = maxgoldi,j−1, ξi,j−1
.
The projected SOR method is used to solve the linear system.1 At each time
j, we iteratively solve
g(k+1)1,j−1 = max
ξ1,j−1 , g
(k)1,j−1 +
ω
1− β1
[r1,j − (1− β1)g(k)1,j−1 + γ1g
(k)2,j−1]
,
g(k+1)2,j−1 = max
ξ2,j−1 , g
(k)2,j−1 +
ω
1− β2
[r2,j + α2g(k+1)1,j−1 − (1− β2)g
(k)2,j−1 + γ2g
(k)3,j−1]
,
...
g(k+1)M−1,j−1 = max
ξM−1,j−1 , g
(k)M−1,j−1
+ω
1− βM−1
[rM−1,j + αM−1g(k+1)M−2,j−1 − (1− βM−1)g
(k)M−1,j−1]
,
where k is the iteration counter and ω is the overrelaxation parameter. The
iterative scheme starts from an initial point g(0)j and proceeds until a conver-
gence criterion is met, such as ||g(k+1)j−1 − g
(k)j−1|| < ε, where ε is a tolerance
parameter. The optimal boundary Sf (t) can be identified by locating the
boundary that separates the regions where g(t, s) = ξ(t, s), or g(t, s) ≥ ξ(t, s).
1For a detailed discussion on the projected SOR method, we refer to Wilmott et al.
[1995].
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